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Properties

Label 4-2312e2-1.1-c1e2-0-9
Degree $4$
Conductor $5345344$
Sign $1$
Analytic cond. $340.823$
Root an. cond. $4.29667$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·9-s + 8·13-s + 16·19-s − 8·25-s + 16·43-s − 6·49-s − 24·53-s + 16·59-s + 24·67-s − 5·81-s + 16·89-s − 24·101-s + 16·103-s + 16·117-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 32·171-s + 173-s + ⋯
L(s)  = 1  + 2/3·9-s + 2.21·13-s + 3.67·19-s − 8/5·25-s + 2.43·43-s − 6/7·49-s − 3.29·53-s + 2.08·59-s + 2.93·67-s − 5/9·81-s + 1.69·89-s − 2.38·101-s + 1.57·103-s + 1.47·117-s − 1.27·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 2.44·171-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5345344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5345344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5345344\)    =    \(2^{6} \cdot 17^{4}\)
Sign: $1$
Analytic conductor: \(340.823\)
Root analytic conductor: \(4.29667\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5345344,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.887303169\)
\(L(\frac12)\) \(\approx\) \(3.887303169\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17 \( 1 \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.5.a_i
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.13.ai_bq
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.19.aq_dy
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.23.a_bm
29$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.29.a_bo
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.31.a_ak
37$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \) 2.37.a_y
41$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \) 2.41.a_dc
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.43.aq_fu
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.53.y_jq
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.59.aq_ha
61$C_2^2$ \( 1 + 120 T^{2} + p^{2} T^{4} \) 2.61.a_eq
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.71.a_cs
73$C_2^2$ \( 1 + 128 T^{2} + p^{2} T^{4} \) 2.73.a_ey
79$C_2^2$ \( 1 + 150 T^{2} + p^{2} T^{4} \) 2.79.a_fu
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.89.aq_ji
97$C_2^2$ \( 1 + 176 T^{2} + p^{2} T^{4} \) 2.97.a_gu
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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

−9.114983642853817129541141289229, −8.995775266392768953121788317049, −8.326211835987934304350881964533, −7.937385241716098415572168880703, −7.64117006881790312529037937833, −7.50404280318145860229621803048, −6.79714982768701302967884710978, −6.52737485143285882584087907882, −5.96892489277558984357441023089, −5.75158071653231284803528873585, −5.22065436497575766249670702703, −4.99936432819021215929049381172, −4.23777283923172370488685199031, −3.72233758233307315164987906687, −3.62404890751513656577975573578, −3.11559129153771067718257422577, −2.48510885410420285412813338199, −1.68106222694686911491000137785, −1.21304116967487993763820066742, −0.806873827848139237358564234971, 0.806873827848139237358564234971, 1.21304116967487993763820066742, 1.68106222694686911491000137785, 2.48510885410420285412813338199, 3.11559129153771067718257422577, 3.62404890751513656577975573578, 3.72233758233307315164987906687, 4.23777283923172370488685199031, 4.99936432819021215929049381172, 5.22065436497575766249670702703, 5.75158071653231284803528873585, 5.96892489277558984357441023089, 6.52737485143285882584087907882, 6.79714982768701302967884710978, 7.50404280318145860229621803048, 7.64117006881790312529037937833, 7.937385241716098415572168880703, 8.326211835987934304350881964533, 8.995775266392768953121788317049, 9.114983642853817129541141289229

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