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Properties

Label 2-845-13.9-c1-0-42
Degree $2$
Conductor $845$
Sign $-0.664 + 0.746i$
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  + (0.747 − 1.29i)2-s + (−0.0473 + 0.0820i)3-s + (−0.118 − 0.204i)4-s − 5-s + (0.0708 + 0.122i)6-s + (−2.41 − 4.18i)7-s + 2.63·8-s + (1.49 + 2.59i)9-s + (−0.747 + 1.29i)10-s + (0.534 − 0.926i)11-s + 0.0224·12-s − 7.21·14-s + (0.0473 − 0.0820i)15-s + (2.20 − 3.82i)16-s + (−1.77 − 3.08i)17-s + 4.47·18-s + ⋯
L(s)  = 1  + (0.528 − 0.915i)2-s + (−0.0273 + 0.0473i)3-s + (−0.0591 − 0.102i)4-s − 0.447·5-s + (0.0289 + 0.0501i)6-s + (−0.912 − 1.57i)7-s + 0.932·8-s + (0.498 + 0.863i)9-s + (−0.236 + 0.409i)10-s + (0.161 − 0.279i)11-s + 0.00647·12-s − 1.92·14-s + (0.0122 − 0.0211i)15-s + (0.552 − 0.956i)16-s + (−0.431 − 0.747i)17-s + 1.05·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 + 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.664 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.664 + 0.746i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.664 + 0.746i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.708114 - 1.57847i\)
\(L(\frac12)\) \(\approx\) \(0.708114 - 1.57847i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
13 \( 1 \)
good2 \( 1 + (-0.747 + 1.29i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.0473 - 0.0820i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (2.41 + 4.18i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.534 + 0.926i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.77 + 3.08i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.86 + 4.96i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.54 + 6.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.736 - 1.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.46T + 31T^{2} \)
37 \( 1 + (-0.0126 + 0.0219i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.133 - 0.232i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.77 + 3.08i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.51T + 47T^{2} \)
53 \( 1 - 0.991T + 53T^{2} \)
59 \( 1 + (4.36 + 7.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.16 + 5.48i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.58 - 4.48i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.88 - 6.72i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 - 8.78T + 79T^{2} \)
83 \( 1 + 0.725T + 83T^{2} \)
89 \( 1 + (6.75 - 11.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.71 - 2.97i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.31885045804464693497015154145, −9.287893691814625267466331615660, −8.072286575832415679876346413152, −7.10365137130797223270291180692, −6.77860838695208586113891025086, −4.87652630389837023098961067091, −4.30437528437193980749923665895, −3.43816638033658784201803179639, −2.41110338092825419507978399886, −0.72360094723884679659319628720, 1.80008660751092900441386295726, 3.36297355758466338405132236801, 4.30431231964684314842256222343, 5.56012386456583112452964924578, 6.12703767372872699904383523927, 6.80317567621797898968881347230, 7.75017867606937020077034453451, 8.785733816743036635999145971431, 9.489135260403753079089726457099, 10.39283408855286840768475887671

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