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Properties

Label 2-867-51.44-c1-0-8
Degree $2$
Conductor $867$
Sign $-0.972 + 0.231i$
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 + (−1.33 + 1.10i)3-s + (1.51 + 1.51i)4-s + (0.405 + 2.03i)5-s + (−3.37 + 1.03i)6-s + (−1.95 − 0.388i)7-s + (0.112 + 0.272i)8-s + (0.565 − 2.94i)9-s + (−0.825 + 4.14i)10-s + (−1.95 + 1.30i)11-s + (−3.69 − 0.351i)12-s + (−4.24 + 4.24i)13-s + (−3.37 − 2.25i)14-s + (−2.78 − 2.27i)15-s − 3.68i·16-s + ⋯
L(s)  = 1  + (1.33 + 0.550i)2-s + (−0.770 + 0.636i)3-s + (0.758 + 0.758i)4-s + (0.181 + 0.911i)5-s + (−1.37 + 0.422i)6-s + (−0.738 − 0.146i)7-s + (0.0399 + 0.0964i)8-s + (0.188 − 0.982i)9-s + (−0.260 + 1.31i)10-s + (−0.590 + 0.394i)11-s + (−1.06 − 0.101i)12-s + (−1.17 + 1.17i)13-s + (−0.901 − 0.602i)14-s + (−0.720 − 0.586i)15-s − 0.922i·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $-0.972 + 0.231i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ -0.972 + 0.231i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.156248 - 1.32895i\)
\(L(\frac12)\) \(\approx\) \(0.156248 - 1.32895i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−10.55465960177271458287233893416, −9.982532628187276860943182735295, −9.239685065492919999128147789227, −7.48415899304046521110120855604, −6.76085441344148853619756030792, −6.24952486530907067995964812519, −5.31934572047860014343323734164, −4.45998854395654506275755333639, −3.66982440048577505320397609038, −2.56843193062669678838428885185, 0.42509877905795382459291173832, 2.19598906019714179822763272422, 3.11281166951507708886721322048, 4.57713763012448561844590997957, 5.25300653970834619539879508427, 5.76255367504793947271175739152, 6.81106670500152490441136649906, 7.87211782284969661676524867234, 8.865066588577798393728906823851, 10.05283434512224052973371424514

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