↑

Properties

Label 2-845-13.10-c1-0-1
Degree $2$
Conductor $845$
Sign $-0.890 - 0.454i$
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  + (0.358 − 0.207i)2-s + (−0.707 − 1.22i)3-s + (−0.914 + 1.58i)4-s + i·5-s + (−0.507 − 0.292i)6-s + (−0.717 − 0.414i)7-s + 1.58i·8-s + (0.500 − 0.866i)9-s + (0.207 + 0.358i)10-s + (−0.507 + 0.292i)11-s + 2.58·12-s − 0.343·14-s + (1.22 − 0.707i)15-s + (−1.49 − 2.59i)16-s + (−2.41 + 4.18i)17-s − 0.414i·18-s + ⋯
L(s)  = 1  + (0.253 − 0.146i)2-s + (−0.408 − 0.707i)3-s + (−0.457 + 0.791i)4-s + 0.447i·5-s + (−0.207 − 0.119i)6-s + (−0.271 − 0.156i)7-s + 0.560i·8-s + (0.166 − 0.288i)9-s + (0.0654 + 0.113i)10-s + (−0.152 + 0.0883i)11-s + 0.746·12-s − 0.0917·14-s + (0.316 − 0.182i)15-s + (−0.374 − 0.649i)16-s + (−0.585 + 1.01i)17-s − 0.0976i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.890 - 0.454i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.890 - 0.454i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0516160 + 0.214627i\)
\(L(\frac12)\) \(\approx\) \(0.0516160 + 0.214627i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - iT \)
13 \( 1 \)
good2 \( 1 + (-0.358 + 0.207i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.707 + 1.22i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.717 + 0.414i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.507 - 0.292i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.41 - 4.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.95 + 1.70i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.707 + 1.22i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.82 + 4.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.2iT - 31T^{2} \)
37 \( 1 + (7.34 - 4.24i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.64 - 4.41i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.53 + 2.65i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.828iT - 47T^{2} \)
53 \( 1 + 14.4T + 53T^{2} \)
59 \( 1 + (-8.87 - 5.12i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.73 - i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.84 - 3.94i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 8.48iT - 73T^{2} \)
79 \( 1 - 8.48T + 79T^{2} \)
83 \( 1 + 8.82iT - 83T^{2} \)
89 \( 1 + (5.19 - 3i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.16 + 1.82i)T + (48.5 + 84.0i)T^{2} \)
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

−10.67726016482488144973895443175, −9.763132822706757706431501041714, −8.671291153592157444380304934579, −8.028480526386806112912604637833, −6.86752714848649559881761224105, −6.54792799352244678673927999751, −5.22444931910505936280755095742, −4.10377720774334137788015289147, −3.25153237710162784189544528081, −1.90721386190526664412952252915, 0.10031373292876584528463803537, 1.93692442747115118423361045731, 3.72779827285613495895278900577, 4.63236868484152986447978608815, 5.27163417895177109148706421664, 6.03400993154576514050484558221, 7.10528889944962441541608113129, 8.285964455535037962914632908044, 9.362291075547765231361071957312, 9.675218894516918611973116156603

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