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Properties

Label 2-3236-3236.319-c0-0-0
Degree $2$
Conductor $3236$
Sign $0.932 + 0.361i$
Analytic cond. $1.61497$
Root an. cond. $1.27081$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.108 + 0.994i)2-s + (−0.976 − 0.216i)4-s + (0.752 − 1.83i)5-s + (0.320 − 0.947i)8-s + (0.969 − 0.246i)9-s + (1.74 + 0.947i)10-s + (0.151 − 0.234i)13-s + (0.906 + 0.421i)16-s + (0.00433 + 0.0307i)17-s + (0.139 + 0.990i)18-s + (−1.13 + 1.63i)20-s + (−2.09 − 2.06i)25-s + (0.216 + 0.176i)26-s + (0.934 + 1.65i)29-s + (−0.517 + 0.855i)32-s + ⋯
L(s)  = 1  + (−0.108 + 0.994i)2-s + (−0.976 − 0.216i)4-s + (0.752 − 1.83i)5-s + (0.320 − 0.947i)8-s + (0.969 − 0.246i)9-s + (1.74 + 0.947i)10-s + (0.151 − 0.234i)13-s + (0.906 + 0.421i)16-s + (0.00433 + 0.0307i)17-s + (0.139 + 0.990i)18-s + (−1.13 + 1.63i)20-s + (−2.09 − 2.06i)25-s + (0.216 + 0.176i)26-s + (0.934 + 1.65i)29-s + (−0.517 + 0.855i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3236 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3236 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3236\)    =    \(2^{2} \cdot 809\)
Sign: $0.932 + 0.361i$
Analytic conductor: \(1.61497\)
Root analytic conductor: \(1.27081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3236} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3236,\ (\ :0),\ 0.932 + 0.361i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.311236177\)
\(L(\frac12)\) \(\approx\) \(1.311236177\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.108 - 0.994i)T \)
809 \( 1 + (-0.544 - 0.838i)T \)
good3 \( 1 + (-0.969 + 0.246i)T^{2} \)
5 \( 1 + (-0.752 + 1.83i)T + (-0.712 - 0.701i)T^{2} \)
7 \( 1 + (0.942 + 0.335i)T^{2} \)
11 \( 1 + (-0.906 + 0.421i)T^{2} \)
13 \( 1 + (-0.151 + 0.234i)T + (-0.407 - 0.913i)T^{2} \)
17 \( 1 + (-0.00433 - 0.0307i)T + (-0.961 + 0.276i)T^{2} \)
19 \( 1 + (-0.734 + 0.679i)T^{2} \)
23 \( 1 + (-0.774 + 0.632i)T^{2} \)
29 \( 1 + (-0.934 - 1.65i)T + (-0.517 + 0.855i)T^{2} \)
31 \( 1 + (0.0466 + 0.998i)T^{2} \)
37 \( 1 + (-0.862 - 0.468i)T + (0.544 + 0.838i)T^{2} \)
41 \( 1 + (1.38 + 1.27i)T + (0.0776 + 0.996i)T^{2} \)
43 \( 1 + (0.170 + 0.985i)T^{2} \)
47 \( 1 + (0.291 + 0.956i)T^{2} \)
53 \( 1 + (-0.0200 + 1.28i)T + (-0.999 - 0.0310i)T^{2} \)
59 \( 1 + (0.667 - 0.744i)T^{2} \)
61 \( 1 + (-0.0216 - 0.153i)T + (-0.961 + 0.276i)T^{2} \)
67 \( 1 + (0.830 + 0.557i)T^{2} \)
71 \( 1 + (0.350 + 0.936i)T^{2} \)
73 \( 1 + (0.333 - 1.62i)T + (-0.919 - 0.393i)T^{2} \)
79 \( 1 + (0.170 + 0.985i)T^{2} \)
83 \( 1 + (0.620 - 0.784i)T^{2} \)
89 \( 1 + (-0.647 + 0.998i)T + (-0.407 - 0.913i)T^{2} \)
97 \( 1 + (-0.574 - 0.727i)T + (-0.231 + 0.972i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.492413098898482245646956707640, −8.404854686253996367977605813075, −7.24489499422694244749220712525, −6.53670462676541666471158046830, −5.69549611630757833774186884576, −5.00278572584090151031809456681, −4.56078656293013284065729987756, −3.59289194344427387632999975789, −1.74130933387187488870853931288, −0.923255571745319289851987944646, 1.55586000379204277245782702636, 2.39578435081344552264802637582, 3.08609306562120399051529435941, 3.98496998913821535832684858220, 4.82992171441768845713217991759, 6.01412894745541876275452225722, 6.55480486592284084063518921826, 7.49138114633132521011061723984, 8.054324270535657818717595373481, 9.307526209960521882437943864994

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