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Properties

Label 2-845-65.4-c1-0-19
Degree $2$
Conductor $845$
Sign $0.602 - 0.798i$
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  + (−1.33 + 2.31i)2-s + (0.416 + 0.240i)3-s + (−2.57 − 4.46i)4-s + (−1.48 + 1.67i)5-s + (−1.11 + 0.643i)6-s + (0.403 + 0.698i)7-s + 8.44·8-s + (−1.38 − 2.39i)9-s + (−1.89 − 5.67i)10-s + (−3.18 − 1.83i)11-s − 2.48i·12-s − 2.15·14-s + (−1.02 + 0.341i)15-s + (−6.13 + 10.6i)16-s + (−1.16 + 0.675i)17-s + 7.40·18-s + ⋯
L(s)  = 1  + (−0.945 + 1.63i)2-s + (0.240 + 0.138i)3-s + (−1.28 − 2.23i)4-s + (−0.662 + 0.749i)5-s + (−0.455 + 0.262i)6-s + (0.152 + 0.263i)7-s + 2.98·8-s + (−0.461 − 0.799i)9-s + (−0.600 − 1.79i)10-s + (−0.959 − 0.554i)11-s − 0.716i·12-s − 0.576·14-s + (−0.263 + 0.0882i)15-s + (−1.53 + 2.65i)16-s + (−0.283 + 0.163i)17-s + 1.74·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 - 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.602 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.602 - 0.798i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (654, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.602 - 0.798i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.538936 + 0.268356i\)
\(L(\frac12)\) \(\approx\) \(0.538936 + 0.268356i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.48 - 1.67i)T \)
13 \( 1 \)
good2 \( 1 + (1.33 - 2.31i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.416 - 0.240i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-0.403 - 0.698i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.18 + 1.83i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.16 - 0.675i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.45 + 0.837i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.61 - 3.24i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.20 + 2.09i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 5.28iT - 31T^{2} \)
37 \( 1 + (-1.88 + 3.26i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-7.19 - 4.15i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.88 + 3.39i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.19T + 47T^{2} \)
53 \( 1 - 5.73iT - 53T^{2} \)
59 \( 1 + (-5.18 + 2.99i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.884 - 1.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.94 - 8.56i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.41 + 4.28i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 11.7T + 73T^{2} \)
79 \( 1 - 2.26T + 79T^{2} \)
83 \( 1 + 3.84T + 83T^{2} \)
89 \( 1 + (2.40 + 1.38i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.936 - 1.62i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.984470223662198423651896803823, −9.156683721755719065171039998216, −8.489855851800412209380741228314, −7.72940106793036440379329323041, −7.11237627595891902027677943055, −6.12481815813982617995616026300, −5.49519647547085569361343025546, −4.22197214049752312516932479454, −2.82233496253710463910651867823, −0.51016715159850527574386368823, 1.00331680381374674875877548940, 2.31152212972794523392284221340, 3.19386356985933228836906482217, 4.45068752368327551526828217604, 5.09586214623607479179820169415, 7.31112755858382673688311580716, 7.86878269950009488481305362475, 8.615457536399973200514390475303, 9.197296692191695119712285708443, 10.25483659430336112101377961132

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