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Properties

Label 2-845-13.12-c1-0-12
Degree $2$
Conductor $845$
Sign $-0.969 - 0.246i$
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  + 1.24i·2-s − 0.198·3-s + 0.445·4-s + i·5-s − 0.246i·6-s + 0.198i·7-s + 3.04i·8-s − 2.96·9-s − 1.24·10-s + 4.04i·11-s − 0.0881·12-s − 0.246·14-s − 0.198i·15-s − 2.91·16-s + 5.40·17-s − 3.69i·18-s + ⋯
L(s)  = 1  + 0.881i·2-s − 0.114·3-s + 0.222·4-s + 0.447i·5-s − 0.100i·6-s + 0.0748i·7-s + 1.07i·8-s − 0.986·9-s − 0.394·10-s + 1.22i·11-s − 0.0254·12-s − 0.0660·14-s − 0.0511i·15-s − 0.727·16-s + 1.31·17-s − 0.870i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.969 - 0.246i$
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: \((2,\ 845,\ (\ :1/2),\ -0.969 - 0.246i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.162295 + 1.29450i\)
\(L(\frac12)\) \(\approx\) \(0.162295 + 1.29450i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.24iT - 2T^{2} \)
3 \( 1 + 0.198T + 3T^{2} \)
7 \( 1 - 0.198iT - 7T^{2} \)
11 \( 1 - 4.04iT - 11T^{2} \)
17 \( 1 - 5.40T + 17T^{2} \)
19 \( 1 + 2.18iT - 19T^{2} \)
23 \( 1 + 7.23T + 23T^{2} \)
29 \( 1 + 7.07T + 29T^{2} \)
31 \( 1 + 0.0217iT - 31T^{2} \)
37 \( 1 - 1.49iT - 37T^{2} \)
41 \( 1 - 10.9iT - 41T^{2} \)
43 \( 1 - 7.55T + 43T^{2} \)
47 \( 1 - 1.03iT - 47T^{2} \)
53 \( 1 + 0.554T + 53T^{2} \)
59 \( 1 - 3.89iT - 59T^{2} \)
61 \( 1 + 5.55T + 61T^{2} \)
67 \( 1 - 5.67iT - 67T^{2} \)
71 \( 1 - 9.52iT - 71T^{2} \)
73 \( 1 + 11.5iT - 73T^{2} \)
79 \( 1 + 15.8T + 79T^{2} \)
83 \( 1 + 17.3iT - 83T^{2} \)
89 \( 1 + 3.05iT - 89T^{2} \)
97 \( 1 + 5.02iT - 97T^{2} \)
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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

−10.52007265431427290883288811897, −9.730962316088352124467373590321, −8.696321564712553545762139370522, −7.67855710391389875062000940187, −7.33157224447087958773882207680, −6.11970839860899619708764159044, −5.70418290650293199729120014120, −4.54603385692484443699563048003, −3.09155909457486913056343393589, −2.02431824320083092752113002932, 0.60884228691609025056961329716, 2.02906061401676048334999734292, 3.25504997469468640789351320769, 3.93937639531506961987706567725, 5.67438786895648331391453968479, 5.90875784599691036174263866051, 7.37975152358243053332033309040, 8.214147855426869825138750799351, 9.101741612941747803214686372543, 9.991200618711590459835113048543

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