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Properties

Label 2-845-13.4-c1-0-46
Degree $2$
Conductor $845$
Sign $0.252 + 0.967i$
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  + (1.40 + 0.809i)2-s + (1.11 − 1.93i)3-s + (0.309 + 0.535i)4-s − i·5-s + (3.13 − 1.80i)6-s + (−0.204 + 0.118i)7-s − 2.23i·8-s + (−1 − 1.73i)9-s + (0.809 − 1.40i)10-s + (−3.66 − 2.11i)11-s + 1.38·12-s − 0.381·14-s + (−1.93 − 1.11i)15-s + (2.42 − 4.20i)16-s + (2.73 + 4.73i)17-s − 3.23i·18-s + ⋯
L(s)  = 1  + (0.990 + 0.572i)2-s + (0.645 − 1.11i)3-s + (0.154 + 0.267i)4-s − 0.447i·5-s + (1.27 − 0.738i)6-s + (−0.0772 + 0.0446i)7-s − 0.790i·8-s + (−0.333 − 0.577i)9-s + (0.255 − 0.443i)10-s + (−1.10 − 0.638i)11-s + 0.398·12-s − 0.102·14-s + (−0.500 − 0.288i)15-s + (0.606 − 1.05i)16-s + (0.663 + 1.14i)17-s − 0.762i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.252 + 0.967i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.252 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.27993 - 1.76109i\)
\(L(\frac12)\) \(\approx\) \(2.27993 - 1.76109i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + iT \)
13 \( 1 \)
good2 \( 1 + (-1.40 - 0.809i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.11 + 1.93i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (0.204 - 0.118i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.66 + 2.11i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.73 - 4.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.204 + 0.118i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.11 + 7.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.736 - 1.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (2.59 + 1.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.14 + 2.97i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.881 + 1.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.9iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-11.0 + 6.35i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.20 - 10.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.27 - 5.35i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.52 + 0.881i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 8.94iT - 83T^{2} \)
89 \( 1 + (-7.79 - 4.5i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.73 - 2.73i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.04655515986453410464705666351, −8.758808777605134656882114814777, −8.180681397719358395470230194217, −7.33350122874357144054893605298, −6.50268685263644860346355559292, −5.65477586102029003742329990490, −4.83044246205459270685806532923, −3.59797526781700916858178816829, −2.52189268742950821639140934058, −0.998290877453678516090630527421, 2.28983723395628549839684868102, 3.22493365576754116670064688257, 3.74406532181029156150727558120, 5.04654446003982046253997974471, 5.24735483970006639514864939916, 6.96966954112005768666111952796, 7.892112143001541075968431266521, 8.835209895244128679422092506707, 9.890465046010624468204828382587, 10.20911551740092878529554599580

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