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Properties

Label 2-867-51.5-c1-0-2
Degree $2$
Conductor $867$
Sign $-0.900 - 0.434i$
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  + (−1.88 + 0.779i)2-s + (−1.10 + 1.33i)3-s + (1.51 − 1.51i)4-s + (−2.03 − 0.405i)5-s + (1.03 − 3.37i)6-s + (−0.388 − 1.95i)7-s + (−0.112 + 0.272i)8-s + (−0.565 − 2.94i)9-s + (4.14 − 0.825i)10-s + (1.30 − 1.95i)11-s + (0.351 + 3.69i)12-s + (−4.24 − 4.24i)13-s + (2.25 + 3.37i)14-s + (2.78 − 2.27i)15-s + 3.68i·16-s + ⋯
L(s)  = 1  + (−1.33 + 0.550i)2-s + (−0.636 + 0.770i)3-s + (0.758 − 0.758i)4-s + (−0.911 − 0.181i)5-s + (0.422 − 1.37i)6-s + (−0.146 − 0.738i)7-s + (−0.0399 + 0.0964i)8-s + (−0.188 − 0.982i)9-s + (1.31 − 0.260i)10-s + (0.394 − 0.590i)11-s + (0.101 + 1.06i)12-s + (−1.17 − 1.17i)13-s + (0.602 + 0.901i)14-s + (0.720 − 0.586i)15-s + 0.922i·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $-0.900 - 0.434i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ -0.900 - 0.434i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0302625 + 0.132216i\)
\(L(\frac12)\) \(\approx\) \(0.0302625 + 0.132216i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.10 - 1.33i)T \)
17 \( 1 \)
good2 \( 1 + (1.88 - 0.779i)T + (1.41 - 1.41i)T^{2} \)
5 \( 1 + (2.03 + 0.405i)T + (4.61 + 1.91i)T^{2} \)
7 \( 1 + (0.388 + 1.95i)T + (-6.46 + 2.67i)T^{2} \)
11 \( 1 + (-1.30 + 1.95i)T + (-4.20 - 10.1i)T^{2} \)
13 \( 1 + (4.24 + 4.24i)T + 13iT^{2} \)
19 \( 1 + (-1.93 - 4.67i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.734 + 0.491i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (-0.0116 + 0.0587i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + (2.44 - 1.63i)T + (11.8 - 28.6i)T^{2} \)
37 \( 1 + (-1.52 - 2.28i)T + (-14.1 + 34.1i)T^{2} \)
41 \( 1 + (3.25 - 0.647i)T + (37.8 - 15.6i)T^{2} \)
43 \( 1 + (2.53 - 6.12i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + (-5.76 + 5.76i)T - 47iT^{2} \)
53 \( 1 + (-1.75 + 0.725i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (3.54 - 8.56i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (5.30 - 1.05i)T + (56.3 - 23.3i)T^{2} \)
67 \( 1 + 7.09iT - 67T^{2} \)
71 \( 1 + (-0.742 + 0.496i)T + (27.1 - 65.5i)T^{2} \)
73 \( 1 + (-0.551 + 2.77i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (-3.30 - 2.20i)T + (30.2 + 72.9i)T^{2} \)
83 \( 1 + (-1.44 - 3.49i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-2.00 - 2.00i)T + 89iT^{2} \)
97 \( 1 + (18.4 + 3.67i)T + (89.6 + 37.1i)T^{2} \)
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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

−10.24927501894014233043570810984, −9.803985727784052439454329850011, −8.828419220188257470879599999452, −7.954277849382604182106870494664, −7.41409224179618079797117698085, −6.40987854062143541288560456434, −5.40297702236917504873976808028, −4.20868735270779266065582696810, −3.39568940972209316954278917092, −0.882819456380193893076117812053, 0.15499928434916472677534749735, 1.78342938101328070153602464247, 2.65679371219829441090727209967, 4.41297650467631894080720040762, 5.45319867444807510006702172315, 6.89376733721988522049537476641, 7.28665773030126861699631098766, 8.117745629448971011123032288783, 9.133966423592758295322568157176, 9.618047689133371482355414863862

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