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Properties

Label 2-867-17.13-c1-0-29
Degree $2$
Conductor $867$
Sign $0.992 + 0.122i$
Analytic cond. $6.92302$
Root an. cond. $2.63116$
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  + i·2-s + (−0.707 − 0.707i)3-s + 4-s + (0.707 − 0.707i)6-s + (2.82 − 2.82i)7-s + 3i·8-s + 1.00i·9-s + (2.82 − 2.82i)11-s + (−0.707 − 0.707i)12-s − 2·13-s + (2.82 + 2.82i)14-s − 16-s − 1.00·18-s − 4i·19-s − 4.00·21-s + (2.82 + 2.82i)22-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.408 − 0.408i)3-s + 0.5·4-s + (0.288 − 0.288i)6-s + (1.06 − 1.06i)7-s + 1.06i·8-s + 0.333i·9-s + (0.852 − 0.852i)11-s + (−0.204 − 0.204i)12-s − 0.554·13-s + (0.755 + 0.755i)14-s − 0.250·16-s − 0.235·18-s − 0.917i·19-s − 0.872·21-s + (0.603 + 0.603i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $0.992 + 0.122i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ 0.992 + 0.122i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83571 - 0.112568i\)
\(L(\frac12)\) \(\approx\) \(1.83571 - 0.112568i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 \)
good2 \( 1 - iT - 2T^{2} \)
5 \( 1 + 5iT^{2} \)
7 \( 1 + (-2.82 + 2.82i)T - 7iT^{2} \)
11 \( 1 + (-2.82 + 2.82i)T - 11iT^{2} \)
13 \( 1 + 2T + 13T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (2.82 - 2.82i)T - 23iT^{2} \)
29 \( 1 + 29iT^{2} \)
31 \( 1 + (-2.82 - 2.82i)T + 31iT^{2} \)
37 \( 1 + (5.65 + 5.65i)T + 37iT^{2} \)
41 \( 1 + (-5.65 + 5.65i)T - 41iT^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 12iT - 59T^{2} \)
61 \( 1 + (-5.65 + 5.65i)T - 61iT^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + (8.48 + 8.48i)T + 71iT^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + (2.82 - 2.82i)T - 79iT^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + (-11.3 - 11.3i)T + 97iT^{2} \)
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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

−10.47972103443161936000742620654, −9.047430374565853923986910774092, −8.137147224408448517494664285148, −7.43810866843630982711459425600, −6.82774272324287118620274547717, −5.94583115840277757570483074124, −5.02926382336883831668377067466, −3.99799860331990027746249921808, −2.36943668241269769636834931132, −1.04011391364982053581673139782, 1.53927791480247816916690213492, 2.40105704805085328042931097410, 3.77751178235424578980270167458, 4.74623200455836089600652149511, 5.71789386427427522169938518220, 6.65067942891167034305187767781, 7.63753647549943068065917571428, 8.662146870496778611954806530256, 9.641703368671938121988931489735, 10.18942098605624172458151517602

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