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Properties

Label 2-812-812.447-c0-0-5
Degree $2$
Conductor $812$
Sign $-0.560 + 0.828i$
Analytic cond. $0.405240$
Root an. cond. $0.636585$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s − 1.00i·4-s + 5-s − 1.00·6-s + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s + (0.707 − 0.707i)10-s + (0.707 − 0.707i)11-s + (−0.707 + 0.707i)12-s − 13-s + 1.00i·14-s + (−0.707 − 0.707i)15-s − 1.00·16-s + (1 − i)17-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s − 1.00i·4-s + 5-s − 1.00·6-s + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s + (0.707 − 0.707i)10-s + (0.707 − 0.707i)11-s + (−0.707 + 0.707i)12-s − 13-s + 1.00i·14-s + (−0.707 − 0.707i)15-s − 1.00·16-s + (1 − i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 812 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(812\)    =    \(2^{2} \cdot 7 \cdot 29\)
Sign: $-0.560 + 0.828i$
Analytic conductor: \(0.405240\)
Root analytic conductor: \(0.636585\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{812} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 812,\ (\ :0),\ -0.560 + 0.828i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.185860846\)
\(L(\frac12)\) \(\approx\) \(1.185860846\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 - iT \)
good3 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
5 \( 1 - T + T^{2} \)
11 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 + (-1 + i)T - iT^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 - T^{2} \)
31 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
37 \( 1 + (1 - i)T - iT^{2} \)
41 \( 1 + (-1 - i)T + iT^{2} \)
43 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
47 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 - 1.41T + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (-1 - i)T + iT^{2} \)
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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

−10.09681437149126920971043608558, −9.622521580747363288553250416430, −8.825992507029401808916993490060, −7.06936210023868316045297196296, −6.43384101726823293800780168390, −5.66457160963702274612591847344, −5.12212352678925465463818116964, −3.44528831791808685943812223509, −2.50223217345007271955091930500, −1.16408216204195686297358452815, 2.31772335607991590560388742198, 3.85970809793892810819533966428, 4.50703397517972438636839353799, 5.63344496041064564280497609817, 6.07358667201616527628982980510, 7.10103021722078039993648904651, 7.88662445369547412769383547768, 9.372490254713654281028812051233, 9.861070769290359387387494700874, 10.57855653999425758425935012046

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