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Properties

Label 2-867-17.15-c1-0-7
Degree $2$
Conductor $867$
Sign $0.774 + 0.632i$
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  + (−1.46 − 1.46i)2-s + (−0.382 − 0.923i)3-s + 2.27i·4-s + (−0.321 + 0.133i)5-s + (−0.790 + 1.90i)6-s + (−4.00 − 1.65i)7-s + (0.394 − 0.394i)8-s + (−0.707 + 0.707i)9-s + (0.663 + 0.275i)10-s + (0.279 − 0.674i)11-s + (2.09 − 0.868i)12-s + 5.40i·13-s + (3.42 + 8.27i)14-s + (0.245 + 0.245i)15-s + 3.38·16-s + ⋯
L(s)  = 1  + (−1.03 − 1.03i)2-s + (−0.220 − 0.533i)3-s + 1.13i·4-s + (−0.143 + 0.0595i)5-s + (−0.322 + 0.779i)6-s + (−1.51 − 0.627i)7-s + (0.139 − 0.139i)8-s + (−0.235 + 0.235i)9-s + (0.209 + 0.0869i)10-s + (0.0842 − 0.203i)11-s + (0.605 − 0.250i)12-s + 1.49i·13-s + (0.916 + 2.21i)14-s + (0.0634 + 0.0634i)15-s + 0.846·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $0.774 + 0.632i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (712, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ 0.774 + 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.403669 - 0.143840i\)
\(L(\frac12)\) \(\approx\) \(0.403669 - 0.143840i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.382 + 0.923i)T \)
17 \( 1 \)
good2 \( 1 + (1.46 + 1.46i)T + 2iT^{2} \)
5 \( 1 + (0.321 - 0.133i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (4.00 + 1.65i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.279 + 0.674i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 - 5.40iT - 13T^{2} \)
19 \( 1 + (2.31 + 2.31i)T + 19iT^{2} \)
23 \( 1 + (-1.36 + 3.28i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-4.04 + 1.67i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-1.26 - 3.05i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (1.09 + 2.65i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-10.9 - 4.53i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (2.64 - 2.64i)T - 43iT^{2} \)
47 \( 1 + 0.476iT - 47T^{2} \)
53 \( 1 + (-7.16 - 7.16i)T + 53iT^{2} \)
59 \( 1 + (-3.74 + 3.74i)T - 59iT^{2} \)
61 \( 1 + (3.27 + 1.35i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 - 1.55T + 67T^{2} \)
71 \( 1 + (3.42 + 8.27i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (3.78 - 1.56i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (0.310 - 0.750i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-5.10 - 5.10i)T + 83iT^{2} \)
89 \( 1 + 9.77iT - 89T^{2} \)
97 \( 1 + (-1.63 + 0.678i)T + (68.5 - 68.5i)T^{2} \)
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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

−10.07574596286911245565268945546, −9.306432605080160347796938987294, −8.767796165778462571071289136406, −7.58873566185648324411214480545, −6.73946742957843929919344680599, −6.07610491922917907631549153186, −4.35331992209104907693451965927, −3.25464770589434107150296974535, −2.24382885063465750733701324784, −0.843912921028561844930910122390, 0.42802864131532338185881788425, 2.84556176495410862184587990313, 3.85919812527991294756661512392, 5.48932019581566924809506452997, 6.00467761157370021866776909617, 6.82557870438905258803597052904, 7.82138784319194176475863331946, 8.594309790401163558456735034309, 9.351549210802258802598658580188, 10.05316475251912159852460975710

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