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Properties

Label 2-845-169.116-c1-0-44
Degree $2$
Conductor $845$
Sign $0.0288 + 0.999i$
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  + (0.0211 + 0.0239i)2-s + (−0.411 − 1.08i)3-s + (0.240 − 1.98i)4-s + (0.239 + 0.970i)5-s + (0.0172 − 0.0328i)6-s + (3.65 − 2.52i)7-s + (0.105 − 0.0725i)8-s + (1.23 − 1.09i)9-s + (−0.0181 + 0.0262i)10-s + (2.67 − 3.02i)11-s + (−2.25 + 0.555i)12-s + (0.552 + 3.56i)13-s + (0.137 + 0.0339i)14-s + (0.955 − 0.659i)15-s + (−3.87 − 0.955i)16-s + (1.69 + 2.45i)17-s + ⋯
L(s)  = 1  + (0.0149 + 0.0169i)2-s + (−0.237 − 0.626i)3-s + (0.120 − 0.992i)4-s + (0.107 + 0.434i)5-s + (0.00703 − 0.0134i)6-s + (1.38 − 0.953i)7-s + (0.0371 − 0.0256i)8-s + (0.412 − 0.365i)9-s + (−0.00573 + 0.00831i)10-s + (0.807 − 0.911i)11-s + (−0.650 + 0.160i)12-s + (0.153 + 0.988i)13-s + (0.0368 + 0.00907i)14-s + (0.246 − 0.170i)15-s + (−0.969 − 0.238i)16-s + (0.410 + 0.594i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0288 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0288 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.0288 + 0.999i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.0288 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38345 - 1.34409i\)
\(L(\frac12)\) \(\approx\) \(1.38345 - 1.34409i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.239 - 0.970i)T \)
13 \( 1 + (-0.552 - 3.56i)T \)
good2 \( 1 + (-0.0211 - 0.0239i)T + (-0.241 + 1.98i)T^{2} \)
3 \( 1 + (0.411 + 1.08i)T + (-2.24 + 1.98i)T^{2} \)
7 \( 1 + (-3.65 + 2.52i)T + (2.48 - 6.54i)T^{2} \)
11 \( 1 + (-2.67 + 3.02i)T + (-1.32 - 10.9i)T^{2} \)
17 \( 1 + (-1.69 - 2.45i)T + (-6.02 + 15.8i)T^{2} \)
19 \( 1 - 3.87iT - 19T^{2} \)
23 \( 1 - 4.41T + 23T^{2} \)
29 \( 1 + (-1.55 + 1.38i)T + (3.49 - 28.7i)T^{2} \)
31 \( 1 + (2.35 - 4.49i)T + (-17.6 - 25.5i)T^{2} \)
37 \( 1 + (3.38 - 6.44i)T + (-21.0 - 30.4i)T^{2} \)
41 \( 1 + (5.93 - 2.24i)T + (30.6 - 27.1i)T^{2} \)
43 \( 1 + (-9.38 + 4.92i)T + (24.4 - 35.3i)T^{2} \)
47 \( 1 + (-3.05 + 0.370i)T + (45.6 - 11.2i)T^{2} \)
53 \( 1 + (7.77 + 11.2i)T + (-18.7 + 49.5i)T^{2} \)
59 \( 1 + (2.31 + 9.37i)T + (-52.2 + 27.4i)T^{2} \)
61 \( 1 + (6.31 - 9.14i)T + (-21.6 - 57.0i)T^{2} \)
67 \( 1 + (15.5 - 1.88i)T + (65.0 - 16.0i)T^{2} \)
71 \( 1 + (10.1 - 3.84i)T + (53.1 - 47.0i)T^{2} \)
73 \( 1 + (-1.51 + 1.70i)T + (-8.79 - 72.4i)T^{2} \)
79 \( 1 + (-1.46 - 12.0i)T + (-76.7 + 18.9i)T^{2} \)
83 \( 1 + (-3.66 - 1.39i)T + (62.1 + 55.0i)T^{2} \)
89 \( 1 - 0.802iT - 89T^{2} \)
97 \( 1 + (3.37 - 13.6i)T + (-85.8 - 45.0i)T^{2} \)
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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

−10.22120840201538529624255124216, −9.178960554006512279236284971540, −8.214381005356122004622716549051, −7.14922702803628821731447751140, −6.60650632139609368819501965037, −5.77631472043825923916911637385, −4.63665642109499475713557378756, −3.67736436760565956442733430657, −1.64818903101067272409472168334, −1.20286893409564958521822671978, 1.71534236255933874694147639559, 2.93069597990323223439748997473, 4.40229296285068483603639015959, 4.80577841821462930617504505560, 5.77528323319749665695468855214, 7.35497632218014371964191093999, 7.72963177481594256780931731053, 8.982661931768002319332890536039, 9.184182320459599071793179099417, 10.58983112264262429282814671213

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