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Properties

Label 2-845-65.7-c1-0-21
Degree $2$
Conductor $845$
Sign $0.989 - 0.142i$
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  + (0.113 + 0.0656i)2-s + (−0.0890 − 0.332i)3-s + (−0.991 − 1.71i)4-s + (2.08 + 0.813i)5-s + (0.0116 − 0.0436i)6-s + (1.39 + 2.40i)7-s − 0.522i·8-s + (2.49 − 1.44i)9-s + (0.183 + 0.229i)10-s + (1.04 + 3.91i)11-s + (−0.482 + 0.482i)12-s + 0.365i·14-s + (0.0847 − 0.764i)15-s + (−1.94 + 3.37i)16-s + (2.34 + 0.627i)17-s + 0.378·18-s + ⋯
L(s)  = 1  + (0.0804 + 0.0464i)2-s + (−0.0513 − 0.191i)3-s + (−0.495 − 0.858i)4-s + (0.931 + 0.363i)5-s + (0.00477 − 0.0178i)6-s + (0.525 + 0.910i)7-s − 0.184i·8-s + (0.831 − 0.480i)9-s + (0.0580 + 0.0724i)10-s + (0.316 + 1.18i)11-s + (−0.139 + 0.139i)12-s + 0.0976i·14-s + (0.0218 − 0.197i)15-s + (−0.487 + 0.843i)16-s + (0.567 + 0.152i)17-s + 0.0891·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.142i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.989 - 0.142i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.989 - 0.142i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88541 + 0.134816i\)
\(L(\frac12)\) \(\approx\) \(1.88541 + 0.134816i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.08 - 0.813i)T \)
13 \( 1 \)
good2 \( 1 + (-0.113 - 0.0656i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.0890 + 0.332i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.39 - 2.40i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.04 - 3.91i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.34 - 0.627i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.83 + 0.491i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (7.70 - 2.06i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-3.96 - 2.28i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.87 - 3.87i)T + 31iT^{2} \)
37 \( 1 + (-3.50 + 6.07i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.20 + 1.66i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.67 + 6.24i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 0.512T + 47T^{2} \)
53 \( 1 + (1.32 - 1.32i)T - 53iT^{2} \)
59 \( 1 + (0.679 - 2.53i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.641 - 1.11i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.13 + 1.80i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.66 + 6.20i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + 9.93iT - 73T^{2} \)
79 \( 1 + 8.37iT - 79T^{2} \)
83 \( 1 - 3.17T + 83T^{2} \)
89 \( 1 + (6.01 - 1.61i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (10.1 - 5.88i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.07310233859061812701025365811, −9.511421910111506388797782605854, −8.800378168039940777208032780980, −7.53971989753605937851294345060, −6.51503943642436413992641683480, −5.88853349905496892071778732952, −4.99755930379792521766285254722, −4.05967283751629890938257116683, −2.23985670328596479533782074102, −1.45856498846809771401175638751, 1.10444289806714292029363608533, 2.64582985170221863960370322375, 4.08129605018566613761265586288, 4.52170398446535647486764146398, 5.71551157467651672696846289150, 6.69954850673577266854068810681, 8.075813010942802114908552582846, 8.137056593464509927238417286020, 9.490693204031898793603953957498, 10.04508982909926836715196009754

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