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Properties

Label 4-1050e2-1.1-c1e2-0-35
Degree $4$
Conductor $1102500$
Sign $1$
Analytic cond. $70.2963$
Root an. cond. $2.89556$
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  + 3·3-s + 4-s + 2·7-s + 6·9-s + 3·12-s + 12·13-s + 16-s − 6·19-s + 6·21-s + 9·27-s + 2·28-s − 8·31-s + 6·36-s − 16·37-s + 36·39-s + 16·43-s + 3·48-s + 3·49-s + 12·52-s − 18·57-s − 4·61-s + 12·63-s + 64-s − 18·67-s + 14·73-s − 6·76-s − 4·79-s + ⋯
L(s)  = 1  + 1.73·3-s + 1/2·4-s + 0.755·7-s + 2·9-s + 0.866·12-s + 3.32·13-s + 1/4·16-s − 1.37·19-s + 1.30·21-s + 1.73·27-s + 0.377·28-s − 1.43·31-s + 36-s − 2.63·37-s + 5.76·39-s + 2.43·43-s + 0.433·48-s + 3/7·49-s + 1.66·52-s − 2.38·57-s − 0.512·61-s + 1.51·63-s + 1/8·64-s − 2.19·67-s + 1.63·73-s − 0.688·76-s − 0.450·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1102500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(70.2963\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1102500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.324217993\)
\(L(\frac12)\) \(\approx\) \(6.324217993\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 - p T + p T^{2} \)
5 \( 1 \)
7$C_1$ \( ( 1 - T )^{2} \)
good11$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.11.a_ad
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.13.am_ck
17$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.17.a_bh
19$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.19.g_bv
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.31.i_da
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.37.q_fi
41$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.41.a_abn
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.47.a_dm
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.a_dm
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.67.s_ih
71$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.71.a_bq
73$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.73.ao_hn
79$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.79.e_gg
83$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.83.a_bt
89$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.89.a_cf
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.97.au_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

−7.982704647601194275162559588174, −7.967535843941389291937593485802, −7.28202966725007180028492050863, −6.82833712853147272597306626000, −6.46177038230604663817255689879, −5.74372284090584252527668413358, −5.68728163064030395991011176054, −4.71485812600927611791203093272, −4.02569848626692601267512154263, −3.91736185403180141186201303911, −3.39930630284680276801854080586, −2.90932005195183109143742073385, −2.01283634789248302897310187156, −1.76448136119712918970498775230, −1.16422993590034766917798668900, 1.16422993590034766917798668900, 1.76448136119712918970498775230, 2.01283634789248302897310187156, 2.90932005195183109143742073385, 3.39930630284680276801854080586, 3.91736185403180141186201303911, 4.02569848626692601267512154263, 4.71485812600927611791203093272, 5.68728163064030395991011176054, 5.74372284090584252527668413358, 6.46177038230604663817255689879, 6.82833712853147272597306626000, 7.28202966725007180028492050863, 7.967535843941389291937593485802, 7.982704647601194275162559588174

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