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Properties

Label 2-845-65.29-c1-0-8
Degree $2$
Conductor $845$
Sign $-0.162 - 0.986i$
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  + (1.02 − 0.593i)2-s + (0.298 − 0.172i)3-s + (−0.295 + 0.511i)4-s + (−1.44 − 1.71i)5-s + (0.204 − 0.354i)6-s + (−1.75 − 1.01i)7-s + 3.07i·8-s + (−1.44 + 2.49i)9-s + (−2.49 − 0.903i)10-s + (1.94 + 3.36i)11-s + 0.203i·12-s − 2.40·14-s + (−0.725 − 0.262i)15-s + (1.23 + 2.14i)16-s + (−4.71 − 2.72i)17-s + 3.42i·18-s + ⋯
L(s)  = 1  + (0.727 − 0.419i)2-s + (0.172 − 0.0996i)3-s + (−0.147 + 0.255i)4-s + (−0.644 − 0.764i)5-s + (0.0836 − 0.144i)6-s + (−0.664 − 0.383i)7-s + 1.08i·8-s + (−0.480 + 0.831i)9-s + (−0.789 − 0.285i)10-s + (0.585 + 1.01i)11-s + 0.0588i·12-s − 0.644·14-s + (−0.187 − 0.0678i)15-s + (0.308 + 0.535i)16-s + (−1.14 − 0.660i)17-s + 0.806i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.162 - 0.986i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (484, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.162 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.615231 + 0.725073i\)
\(L(\frac12)\) \(\approx\) \(0.615231 + 0.725073i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.44 + 1.71i)T \)
13 \( 1 \)
good2 \( 1 + (-1.02 + 0.593i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.298 + 0.172i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (1.75 + 1.01i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.94 - 3.36i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (4.71 + 2.72i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.94 - 5.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.298 - 0.172i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.18T + 31T^{2} \)
37 \( 1 + (-4.71 + 2.72i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.0902 - 0.156i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.15 - 0.669i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.2iT - 47T^{2} \)
53 \( 1 - 2.42iT - 53T^{2} \)
59 \( 1 + (-3.53 + 6.11i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.38 - 5.85i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.81 - 2.20i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.940 - 1.62i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 8.86iT - 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 + 7.83iT - 83T^{2} \)
89 \( 1 + (6.12 + 10.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.02 - 2.90i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.65117040418473649778875736932, −9.460803327494802251648278059756, −8.724670452234040497102791391075, −7.907530470094238894842404373906, −7.13746027016927973154363323548, −5.84557004810064783507099355609, −4.62914931692538913735133802423, −4.27079362576278521211880381014, −3.14082429397210806417979602773, −1.92659553306154573843315534524, 0.34469167441404592145031893089, 2.74294729018876873706773542249, 3.67420362109502054565241802692, 4.36988213415471209956802371597, 5.80650551088049552580800999673, 6.45577014794520577972446723622, 6.88062937014366250236581826101, 8.428104391927506380538262335565, 9.011975496165161406272413955942, 9.910729900101908228177653843534

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