Properties

Label 2-845-13.10-c1-0-33
Degree $2$
Conductor $845$
Sign $0.967 + 0.252i$
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  + (−1.29 + 0.747i)2-s + (−0.0473 − 0.0820i)3-s + (0.118 − 0.204i)4-s i·5-s + (0.122 + 0.0708i)6-s + (4.18 + 2.41i)7-s − 2.63i·8-s + (1.49 − 2.59i)9-s + (0.747 + 1.29i)10-s + (0.926 − 0.534i)11-s − 0.0224·12-s − 7.21·14-s + (−0.0820 + 0.0473i)15-s + (2.20 + 3.82i)16-s + (1.77 − 3.08i)17-s + 4.47i·18-s + ⋯
L(s)  = 1  + (−0.915 + 0.528i)2-s + (−0.0273 − 0.0473i)3-s + (0.0591 − 0.102i)4-s − 0.447i·5-s + (0.0501 + 0.0289i)6-s + (1.57 + 0.912i)7-s − 0.932i·8-s + (0.498 − 0.863i)9-s + (0.236 + 0.409i)10-s + (0.279 − 0.161i)11-s − 0.00647·12-s − 1.92·14-s + (−0.0211 + 0.0122i)15-s + (0.552 + 0.956i)16-s + (0.431 − 0.747i)17-s + 1.05i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.967 + 0.252i$
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: \((2,\ 845,\ (\ :1/2),\ 0.967 + 0.252i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03375 - 0.132726i\)
\(L(\frac12)\) \(\approx\) \(1.03375 - 0.132726i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.29 - 0.747i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.0473 + 0.0820i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-4.18 - 2.41i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.926 + 0.534i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.77 + 3.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.96 + 2.86i)T + (9.5 + 16.4i)T^{2} \)
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 \( 1 + 1.46iT - 31T^{2} \)
37 \( 1 + (-0.0219 + 0.0126i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.232 + 0.133i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.77 + 3.08i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.51iT - 47T^{2} \)
53 \( 1 - 0.991T + 53T^{2} \)
59 \( 1 + (-7.55 - 4.36i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.16 - 5.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.48 + 2.58i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.72 - 3.88i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.1iT - 73T^{2} \)
79 \( 1 - 8.78T + 79T^{2} \)
83 \( 1 + 0.725iT - 83T^{2} \)
89 \( 1 + (11.6 - 6.75i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.97 - 1.71i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.862249937897033273011028385000, −8.967591662193905578058880869623, −8.586657947729784655667640038793, −7.85087719615188249286732441603, −6.90255542193932481844117410864, −5.97948742169194886376911192381, −4.81392116337144513752694829147, −3.97361344777388104785801931470, −2.19604894370553062122683139451, −0.78384325118826629933454676446, 1.42681899750208307142342359318, 2.02204097091666743621171475506, 3.88503630756856136380133426187, 4.77213501532738912871415041970, 5.76729223345913630138509718090, 7.19087638846557101956728945213, 8.000985988716756314295050319801, 8.321468436276314933046970775077, 9.684314787211653270192208015995, 10.28659706874373253273694740700

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