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Properties

Label 2-867-17.4-c1-0-15
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: $\chi_{867} (616, \cdot )$
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

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

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