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Properties

Label 2-845-13.9-c1-0-28
Degree $2$
Conductor $845$
Sign $0.664 - 0.746i$
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  + (0.609 − 1.05i)2-s + (−1.16 + 2.01i)3-s + (0.256 + 0.443i)4-s + 5-s + (1.42 + 2.46i)6-s + (1.80 + 3.11i)7-s + 3.06·8-s + (−1.21 − 2.11i)9-s + (0.609 − 1.05i)10-s + (2.68 − 4.65i)11-s − 1.19·12-s + 4.39·14-s + (−1.16 + 2.01i)15-s + (1.35 − 2.34i)16-s + (0.565 + 0.980i)17-s − 2.97·18-s + ⋯
L(s)  = 1  + (0.431 − 0.746i)2-s + (−0.673 + 1.16i)3-s + (0.128 + 0.221i)4-s + 0.447·5-s + (0.580 + 1.00i)6-s + (0.680 + 1.17i)7-s + 1.08·8-s + (−0.406 − 0.704i)9-s + (0.192 − 0.334i)10-s + (0.809 − 1.40i)11-s − 0.344·12-s + 1.17·14-s + (−0.301 + 0.521i)15-s + (0.339 − 0.587i)16-s + (0.137 + 0.237i)17-s − 0.701·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 - 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.664 - 0.746i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.664 - 0.746i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96100 + 0.879720i\)
\(L(\frac12)\) \(\approx\) \(1.96100 + 0.879720i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
13 \( 1 \)
good2 \( 1 + (-0.609 + 1.05i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.16 - 2.01i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-1.80 - 3.11i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.68 + 4.65i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.565 - 0.980i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.13 - 1.96i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.94 + 3.37i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.0123 + 0.0214i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.46T + 31T^{2} \)
37 \( 1 + (4.35 - 7.53i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.86 + 3.23i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.565 - 0.980i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.58T + 47T^{2} \)
53 \( 1 + 4.43T + 53T^{2} \)
59 \( 1 + (0.0857 + 0.148i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.68 + 2.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.19 + 5.54i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.39 + 9.35i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.70T + 73T^{2} \)
79 \( 1 + 11.9T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 + (8.07 - 13.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.08 - 10.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.73122192386195676827205473871, −9.632546257013516428733655399346, −8.824747787006173835724895781461, −8.030999114057250603967206853924, −6.48748829030025526129875746401, −5.57717250914804363721917138575, −4.94187778006346245328341778709, −3.88537009694859947603213335699, −3.01123355536095690793032785882, −1.67002722809890081136681214572, 1.17730105056341723671484210050, 1.87377430814152765989430284289, 4.08934130050012725805770056991, 5.00272765370528268542925052338, 5.81976953194234687206472772494, 6.88708197171669483305212771211, 7.14064877028083538329558708151, 7.72822937599521302445570604163, 9.287037065071710729683137167027, 10.18507902678104212171431632084

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