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Properties

Label 2-845-65.37-c1-0-7
Degree $2$
Conductor $845$
Sign $-0.803 - 0.594i$
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  + (−2.29 + 1.32i)2-s + (−1.25 − 0.335i)3-s + (2.51 − 4.34i)4-s + (−1.81 + 1.30i)5-s + (3.31 − 0.889i)6-s + (−0.0561 + 0.0972i)7-s + 8.00i·8-s + (−1.14 − 0.658i)9-s + (2.44 − 5.39i)10-s + (1.78 + 0.479i)11-s + (−4.60 + 4.60i)12-s − 0.297i·14-s + (2.71 − 1.02i)15-s + (−5.58 − 9.67i)16-s + (0.706 + 2.63i)17-s + 3.49·18-s + ⋯
L(s)  = 1  + (−1.62 + 0.936i)2-s + (−0.723 − 0.193i)3-s + (1.25 − 2.17i)4-s + (−0.812 + 0.583i)5-s + (1.35 − 0.363i)6-s + (−0.0212 + 0.0367i)7-s + 2.83i·8-s + (−0.380 − 0.219i)9-s + (0.771 − 1.70i)10-s + (0.539 + 0.144i)11-s + (−1.32 + 1.32i)12-s − 0.0795i·14-s + (0.700 − 0.264i)15-s + (−1.39 − 2.41i)16-s + (0.171 + 0.639i)17-s + 0.823·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 - 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.803 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.803 - 0.594i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.803 - 0.594i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0743354 + 0.225419i\)
\(L(\frac12)\) \(\approx\) \(0.0743354 + 0.225419i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.81 - 1.30i)T \)
13 \( 1 \)
good2 \( 1 + (2.29 - 1.32i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (1.25 + 0.335i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.0561 - 0.0972i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.78 - 0.479i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.706 - 2.63i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.80 + 6.72i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.831 + 3.10i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-4.03 + 2.32i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.624 - 0.624i)T + 31iT^{2} \)
37 \( 1 + (-0.737 - 1.27i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.40 - 5.24i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.76 - 1.00i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 - 0.345T + 47T^{2} \)
53 \( 1 + (3.59 - 3.59i)T - 53iT^{2} \)
59 \( 1 + (-1.24 + 0.332i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.39 + 2.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.124 + 0.0721i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.28 - 1.41i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 - 9.06iT - 73T^{2} \)
79 \( 1 - 15.1iT - 79T^{2} \)
83 \( 1 + 8.53T + 83T^{2} \)
89 \( 1 + (-0.147 + 0.549i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (12.9 + 7.48i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.49260401769523422336242428273, −9.544737143642516666363667291324, −8.626201387702127005217789949517, −8.107653756811752401095042200683, −6.90631009232848781580808091516, −6.68930639299355697665711437712, −5.81199198462306149406434553741, −4.54144968521801097750621925681, −2.73784738597263207353321521792, −0.951536001703881398986142017293, 0.29623316250245102333529795452, 1.57582273888687760049923518933, 3.12172882051357462708294590942, 4.10181469932528343446052332121, 5.43661717999672247662189267321, 6.73281974203889649946178523494, 7.70856205524369122945371390272, 8.381809411198200370748457167914, 9.030776627345528831745450274921, 9.969190197398864705494306802198

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