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Properties

Label 4-630e2-1.1-c1e2-0-5
Degree $4$
Conductor $396900$
Sign $1$
Analytic cond. $25.3066$
Root an. cond. $2.24289$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 5-s − 4·7-s + 8-s − 10-s + 3·11-s − 2·13-s + 4·14-s − 16-s − 6·17-s + 19-s − 3·22-s + 9·23-s + 2·26-s − 12·29-s − 8·31-s + 6·34-s − 4·35-s + 7·37-s − 38-s + 40-s − 6·41-s + 4·43-s − 9·46-s + 9·47-s + 9·49-s + 9·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.447·5-s − 1.51·7-s + 0.353·8-s − 0.316·10-s + 0.904·11-s − 0.554·13-s + 1.06·14-s − 1/4·16-s − 1.45·17-s + 0.229·19-s − 0.639·22-s + 1.87·23-s + 0.392·26-s − 2.22·29-s − 1.43·31-s + 1.02·34-s − 0.676·35-s + 1.15·37-s − 0.162·38-s + 0.158·40-s − 0.937·41-s + 0.609·43-s − 1.32·46-s + 1.31·47-s + 9/7·49-s + 1.23·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(396900\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(25.3066\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 396900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7455685926\)
\(L(\frac12)\) \(\approx\) \(0.7455685926\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2$C_2$ \( 1 + T + T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good11$C_2^2$ \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.11.ad_ac
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.13.c_bb
17$C_2^2$ \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.17.g_t
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.ab_as
23$C_2^2$ \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.23.aj_cg
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_bh
37$C_2^2$ \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.37.ah_m
41$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.41.g_dn
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.43.ae_dm
47$C_2^2$ \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.47.aj_bi
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.53.aj_bc
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2^2$ \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_d
67$C_2^2$ \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_ad
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.73.ae_acf
79$C_2^2$ \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.79.ak_v
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_acb
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.97.u_li
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.86037196350447712640105800846, −10.23348268908069349692735258134, −9.810832613109154651187308458365, −9.275038363577339608377732855048, −9.167571816295817541578745753878, −9.027830488649272853844521243567, −8.431322923498462436438461027243, −7.48834055310823392296467258631, −7.22997307536705740678455435116, −7.05508828640405727662233321768, −6.27211740071925058886239990051, −6.09233066087942632195082256806, −5.37344625041636840779741858388, −4.91434467143120060405755887229, −4.06318716330875927530917019269, −3.77740788401466358629796333697, −3.01806832281580732517391860513, −2.38501681317349082141778395324, −1.66147720136269519746512500824, −0.53165783016322973948503440947, 0.53165783016322973948503440947, 1.66147720136269519746512500824, 2.38501681317349082141778395324, 3.01806832281580732517391860513, 3.77740788401466358629796333697, 4.06318716330875927530917019269, 4.91434467143120060405755887229, 5.37344625041636840779741858388, 6.09233066087942632195082256806, 6.27211740071925058886239990051, 7.05508828640405727662233321768, 7.22997307536705740678455435116, 7.48834055310823392296467258631, 8.431322923498462436438461027243, 9.027830488649272853844521243567, 9.167571816295817541578745753878, 9.275038363577339608377732855048, 9.810832613109154651187308458365, 10.23348268908069349692735258134, 10.86037196350447712640105800846

Graph of the $Z$-function along the critical line