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Properties

Label 2-845-169.116-c1-0-50
Degree $2$
Conductor $845$
Sign $0.782 + 0.622i$
Analytic cond. $6.74735$
Root an. cond. $2.59756$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 + 0.196i)2-s + (0.665 + 1.75i)3-s + (0.232 − 1.91i)4-s + (−0.239 − 0.970i)5-s + (−0.228 + 0.435i)6-s + (2.51 − 1.73i)7-s + (0.848 − 0.585i)8-s + (−0.392 + 0.347i)9-s + (0.148 − 0.215i)10-s + (0.754 − 0.851i)11-s + (3.52 − 0.867i)12-s + (−2.99 + 2.00i)13-s + (0.778 + 0.191i)14-s + (1.54 − 1.06i)15-s + (−3.48 − 0.859i)16-s + (−3.60 − 5.22i)17-s + ⋯
L(s)  = 1  + (0.122 + 0.138i)2-s + (0.384 + 1.01i)3-s + (0.116 − 0.958i)4-s + (−0.107 − 0.434i)5-s + (−0.0933 + 0.177i)6-s + (0.951 − 0.656i)7-s + (0.299 − 0.206i)8-s + (−0.130 + 0.115i)9-s + (0.0470 − 0.0682i)10-s + (0.227 − 0.256i)11-s + (1.01 − 0.250i)12-s + (−0.831 + 0.555i)13-s + (0.208 + 0.0512i)14-s + (0.398 − 0.275i)15-s + (−0.872 − 0.214i)16-s + (−0.874 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.782 + 0.622i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.782 + 0.622i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90152 - 0.664394i\)
\(L(\frac12)\) \(\approx\) \(1.90152 - 0.664394i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.239 + 0.970i)T \)
13 \( 1 + (2.99 - 2.00i)T \)
good2 \( 1 + (-0.173 - 0.196i)T + (-0.241 + 1.98i)T^{2} \)
3 \( 1 + (-0.665 - 1.75i)T + (-2.24 + 1.98i)T^{2} \)
7 \( 1 + (-2.51 + 1.73i)T + (2.48 - 6.54i)T^{2} \)
11 \( 1 + (-0.754 + 0.851i)T + (-1.32 - 10.9i)T^{2} \)
17 \( 1 + (3.60 + 5.22i)T + (-6.02 + 15.8i)T^{2} \)
19 \( 1 + 5.33iT - 19T^{2} \)
23 \( 1 - 3.79T + 23T^{2} \)
29 \( 1 + (-0.215 + 0.190i)T + (3.49 - 28.7i)T^{2} \)
31 \( 1 + (4.26 - 8.11i)T + (-17.6 - 25.5i)T^{2} \)
37 \( 1 + (-2.17 + 4.14i)T + (-21.0 - 30.4i)T^{2} \)
41 \( 1 + (0.849 - 0.322i)T + (30.6 - 27.1i)T^{2} \)
43 \( 1 + (-4.20 + 2.20i)T + (24.4 - 35.3i)T^{2} \)
47 \( 1 + (-6.52 + 0.792i)T + (45.6 - 11.2i)T^{2} \)
53 \( 1 + (-0.669 - 0.970i)T + (-18.7 + 49.5i)T^{2} \)
59 \( 1 + (-2.04 - 8.27i)T + (-52.2 + 27.4i)T^{2} \)
61 \( 1 + (3.19 - 4.63i)T + (-21.6 - 57.0i)T^{2} \)
67 \( 1 + (-7.02 + 0.853i)T + (65.0 - 16.0i)T^{2} \)
71 \( 1 + (-11.7 + 4.44i)T + (53.1 - 47.0i)T^{2} \)
73 \( 1 + (5.02 - 5.66i)T + (-8.79 - 72.4i)T^{2} \)
79 \( 1 + (-1.63 - 13.4i)T + (-76.7 + 18.9i)T^{2} \)
83 \( 1 + (-11.2 - 4.28i)T + (62.1 + 55.0i)T^{2} \)
89 \( 1 + 7.12iT - 89T^{2} \)
97 \( 1 + (1.93 - 7.84i)T + (-85.8 - 45.0i)T^{2} \)
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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

−10.04965858403231680169877936596, −9.094085983019356593605609173197, −8.964417854470285558844107519370, −7.31684222474468133085407085582, −6.83979217843561207813606550100, −5.20769166405806669214508169039, −4.79144675507279149946062030791, −4.08820078555076274596421059101, −2.49283621854858395648996019036, −0.950380332620651613621679491441, 1.86133669252305500956732123455, 2.45418995932583939384375456475, 3.75056220088958605244676534628, 4.82327809692820820666274606541, 6.13447065622003667920592349259, 7.08597129072714039036114469537, 7.938831182157459958504925094608, 8.121159833911268066056233835731, 9.198295036536643801860007625855, 10.48829077803627761237545885526

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