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Properties

Label 4-2376e2-1.1-c1e2-0-19
Degree $4$
Conductor $5645376$
Sign $1$
Analytic cond. $359.954$
Root an. cond. $4.35573$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s + 6·25-s − 12·31-s − 6·37-s − 16·47-s − 3·49-s − 8·53-s + 16·59-s + 12·67-s − 16·71-s − 18·97-s − 20·103-s − 12·113-s − 11·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 48·155-s + 157-s + 163-s + 167-s − 22·169-s + 173-s + ⋯
L(s)  = 1  − 1.78·5-s + 6/5·25-s − 2.15·31-s − 0.986·37-s − 2.33·47-s − 3/7·49-s − 1.09·53-s + 2.08·59-s + 1.46·67-s − 1.89·71-s − 1.82·97-s − 1.97·103-s − 1.12·113-s − 121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 3.85·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5645376\)    =    \(2^{6} \cdot 3^{6} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(359.954\)
Root analytic conductor: \(4.35573\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5645376,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
11$C_2$ \( 1 + p T^{2} \)
good5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.e_k
7$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.7.a_d
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2^2$ \( 1 - 23 T^{2} + p^{2} T^{4} \) 2.17.a_ax
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.23.a_bt
29$C_2^2$ \( 1 + 25 T^{2} + p^{2} T^{4} \) 2.29.a_z
31$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.m_dq
37$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.g_t
41$C_2^2$ \( 1 - 55 T^{2} + p^{2} T^{4} \) 2.41.a_acd
43$C_2^2$ \( 1 - 61 T^{2} + p^{2} T^{4} \) 2.43.a_acj
47$C_2$$\times$$C_2$ \( ( 1 + 7 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.47.q_gb
53$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.53.i_ec
59$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.59.aq_gr
61$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.61.a_ak
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.67.am_gk
71$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.q_hi
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.a_bu
79$C_2^2$ \( 1 - 53 T^{2} + p^{2} T^{4} \) 2.79.a_acb
83$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.83.a_cc
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.97.s_id
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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

−6.96879465993623359966806393591, −6.69748955021104135981971576950, −6.04827358917339105420475713580, −5.57571260820188604201792003117, −5.08110441572862246834109997892, −4.89091566704032150293070692440, −4.15092305092204511328110824824, −3.90096173614192286439118852116, −3.55206480118772872746711088458, −3.14373177401014801119395215864, −2.52352406274786635645054599652, −1.78736594070296552885309104120, −1.25794867878740401740310605613, 0, 0, 1.25794867878740401740310605613, 1.78736594070296552885309104120, 2.52352406274786635645054599652, 3.14373177401014801119395215864, 3.55206480118772872746711088458, 3.90096173614192286439118852116, 4.15092305092204511328110824824, 4.89091566704032150293070692440, 5.08110441572862246834109997892, 5.57571260820188604201792003117, 6.04827358917339105420475713580, 6.69748955021104135981971576950, 6.96879465993623359966806393591

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