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Properties

Label 2-867-17.16-c1-0-31
Degree $2$
Conductor $867$
Sign $0.685 - 0.727i$
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  + 2.63·2-s + i·3-s + 4.95·4-s + 2.22i·5-s + 2.63i·6-s − 2.31i·7-s + 7.77·8-s − 9-s + 5.85i·10-s + 1.95i·11-s + 4.95i·12-s + 1.32·13-s − 6.10i·14-s − 2.22·15-s + 10.6·16-s + ⋯
L(s)  = 1  + 1.86·2-s + 0.577i·3-s + 2.47·4-s + 0.993i·5-s + 1.07i·6-s − 0.874i·7-s + 2.75·8-s − 0.333·9-s + 1.85i·10-s + 0.588i·11-s + 1.42i·12-s + 0.366·13-s − 1.63i·14-s − 0.573·15-s + 2.65·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $0.685 - 0.727i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ 0.685 - 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.34743 + 1.87617i\)
\(L(\frac12)\) \(\approx\) \(4.34743 + 1.87617i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
17 \( 1 \)
good2 \( 1 - 2.63T + 2T^{2} \)
5 \( 1 - 2.22iT - 5T^{2} \)
7 \( 1 + 2.31iT - 7T^{2} \)
11 \( 1 - 1.95iT - 11T^{2} \)
13 \( 1 - 1.32T + 13T^{2} \)
19 \( 1 + 5.95T + 19T^{2} \)
23 \( 1 + 4.77iT - 23T^{2} \)
29 \( 1 + 0.172iT - 29T^{2} \)
31 \( 1 + 0.444iT - 31T^{2} \)
37 \( 1 + 5.31iT - 37T^{2} \)
41 \( 1 - 4.02iT - 41T^{2} \)
43 \( 1 + 8.33T + 43T^{2} \)
47 \( 1 - 2.38T + 47T^{2} \)
53 \( 1 - 0.727T + 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 + 8.68iT - 61T^{2} \)
67 \( 1 + 14.1T + 67T^{2} \)
71 \( 1 + 6.10iT - 71T^{2} \)
73 \( 1 - 10.5iT - 73T^{2} \)
79 \( 1 - 0.698iT - 79T^{2} \)
83 \( 1 + 8.52T + 83T^{2} \)
89 \( 1 - 5.40T + 89T^{2} \)
97 \( 1 - 4.71iT - 97T^{2} \)
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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

−10.76118976575850152535442509113, −9.932123390676542934334007441641, −8.356696451943992744839145784234, −7.12212731715851952361923928180, −6.68457215505677066912284891203, −5.80402737022900106655272074849, −4.61568838682495926601784162811, −4.08299106792502571072853873547, −3.17725116547733289198268432635, −2.17208487813861600852531790190, 1.57174299929084077640902227254, 2.67412297434991057915726490836, 3.77714550909312948866633315121, 4.81778004165919955693354671884, 5.59789752704246244541411467749, 6.17591667827402570895589563203, 7.11911983170943477361317058473, 8.308225911064284408535722668025, 8.911553562621244200629866819116, 10.43021050191494070356790206249

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