Properties

Label 2-845-13.12-c1-0-25
Degree $2$
Conductor $845$
Sign $0.960 + 0.277i$
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  − 1.21i·2-s + 2.33·3-s + 0.512·4-s + i·5-s − 2.84i·6-s + 3.60i·7-s − 3.06i·8-s + 2.43·9-s + 1.21·10-s + 5.37i·11-s + 1.19·12-s + 4.39·14-s + 2.33i·15-s − 2.71·16-s + 1.13·17-s − 2.97i·18-s + ⋯
L(s)  = 1  − 0.862i·2-s + 1.34·3-s + 0.256·4-s + 0.447i·5-s − 1.16i·6-s + 1.36i·7-s − 1.08i·8-s + 0.813·9-s + 0.385·10-s + 1.61i·11-s + 0.344·12-s + 1.17·14-s + 0.602i·15-s − 0.678·16-s + 0.274·17-s − 0.701i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.960 + 0.277i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (506, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.960 + 0.277i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.73087 - 0.386280i\)
\(L(\frac12)\) \(\approx\) \(2.73087 - 0.386280i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.21iT - 2T^{2} \)
3 \( 1 - 2.33T + 3T^{2} \)
7 \( 1 - 3.60iT - 7T^{2} \)
11 \( 1 - 5.37iT - 11T^{2} \)
17 \( 1 - 1.13T + 17T^{2} \)
19 \( 1 + 2.26iT - 19T^{2} \)
23 \( 1 - 3.89T + 23T^{2} \)
29 \( 1 + 0.0247T + 29T^{2} \)
31 \( 1 + 5.46iT - 31T^{2} \)
37 \( 1 + 8.70iT - 37T^{2} \)
41 \( 1 + 3.73iT - 41T^{2} \)
43 \( 1 - 1.13T + 43T^{2} \)
47 \( 1 - 2.58iT - 47T^{2} \)
53 \( 1 + 4.43T + 53T^{2} \)
59 \( 1 + 0.171iT - 59T^{2} \)
61 \( 1 - 3.36T + 61T^{2} \)
67 \( 1 + 6.39iT - 67T^{2} \)
71 \( 1 - 10.7iT - 71T^{2} \)
73 \( 1 + 4.70iT - 73T^{2} \)
79 \( 1 + 11.9T + 79T^{2} \)
83 \( 1 - 12.1iT - 83T^{2} \)
89 \( 1 + 16.1iT - 89T^{2} \)
97 \( 1 + 12.1iT - 97T^{2} \)
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

−9.938096825308430814717915965261, −9.432758550968883505391684368936, −8.759132713355858665261823262767, −7.59327638682431634149141234363, −7.01093420883031563655414763344, −5.77294393581941144255023233507, −4.38185973073480649636914480301, −3.25563553938612333802451898929, −2.44188132477319053669433571382, −1.96303641288524439592703328617, 1.29782475790917684072192055650, 2.92739637708683345206152690349, 3.66288863821351000791840156308, 4.94282854353087451321275805029, 6.06730466788803620226640269118, 6.99398678400800700349347429721, 7.82166952220656918976190720563, 8.340808594392782520014199487208, 8.974955121062126399782232581097, 10.12433663312037883527611727966

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