↑

Properties

Label 2-845-65.29-c1-0-4
Degree $2$
Conductor $845$
Sign $-0.866 + 0.498i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.395 − 0.228i)2-s + (−0.866 + 0.5i)3-s + (−0.895 + 1.55i)4-s + (−0.456 + 2.18i)5-s + (−0.228 + 0.395i)6-s + (−1.5 − 0.866i)7-s + 1.73i·8-s + (−1 + 1.73i)9-s + (0.319 + 0.970i)10-s + (1.32 + 2.29i)11-s − 1.79i·12-s − 0.791·14-s + (−0.698 − 2.12i)15-s + (−1.39 − 2.41i)16-s + (−3.96 − 2.29i)17-s + 0.913i·18-s + ⋯
L(s)  = 1  + (0.279 − 0.161i)2-s + (−0.499 + 0.288i)3-s + (−0.447 + 0.775i)4-s + (−0.204 + 0.978i)5-s + (−0.0932 + 0.161i)6-s + (−0.566 − 0.327i)7-s + 0.612i·8-s + (−0.333 + 0.577i)9-s + (0.100 + 0.306i)10-s + (0.398 + 0.690i)11-s − 0.517i·12-s − 0.211·14-s + (−0.180 − 0.548i)15-s + (−0.348 − 0.604i)16-s + (−0.962 − 0.555i)17-s + 0.215i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.866 + 0.498i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (484, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.866 + 0.498i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.113095 - 0.423729i\)
\(L(\frac12)\) \(\approx\) \(0.113095 - 0.423729i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.456 - 2.18i)T \)
13 \( 1 \)
good2 \( 1 + (-0.395 + 0.228i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.866 - 0.5i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (1.5 + 0.866i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.32 - 2.29i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.96 + 2.29i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.866 + 1.5i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.96 + 2.29i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.29 - 3.96i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.20T + 31T^{2} \)
37 \( 1 + (6.87 - 3.96i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.32 - 2.29i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (9.16 + 5.29i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.82iT - 47T^{2} \)
53 \( 1 + 7.58iT - 53T^{2} \)
59 \( 1 + (6.97 - 12.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.708 + 1.22i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.873 - 0.504i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.51 - 6.08i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 - 6.01iT - 83T^{2} \)
89 \( 1 + (4.78 + 8.29i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.87 - 5.70i)T + (48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.70586489082105386036234690771, −10.05637851454931802150278702855, −9.017655295815689218579098769187, −8.134373351105902265920727530783, −7.03181395704569579536758341147, −6.61789376206189329492464367832, −5.10843064828391280114882946145, −4.44125933120349088252253436893, −3.33155712632883462203033370490, −2.49535755684036579207641542211, 0.21839756590916806526859545790, 1.40143656070778313428923789867, 3.38285625969659846949786558759, 4.44758875808949457040712449507, 5.38556626488821674150862496358, 6.13577736877911854760043898950, 6.67674358433400247102031899703, 8.189410099382286111176458721714, 9.041259274582059183643677303502, 9.437135088676028477019290177605

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