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Properties

Label 2-867-17.13-c1-0-0
Degree $2$
Conductor $867$
Sign $0.943 - 0.332i$
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  − 1.84i·2-s + (−0.707 − 0.707i)3-s − 1.41·4-s + (−2.55 − 2.55i)5-s + (−1.30 + 1.30i)6-s + (−2.72 + 2.72i)7-s − 1.08i·8-s + 1.00i·9-s + (−4.72 + 4.72i)10-s + (−0.140 + 0.140i)11-s + (1 + i)12-s + 0.883·13-s + (5.02 + 5.02i)14-s + 3.61i·15-s − 4.82·16-s + ⋯
L(s)  = 1  − 1.30i·2-s + (−0.408 − 0.408i)3-s − 0.707·4-s + (−1.14 − 1.14i)5-s + (−0.533 + 0.533i)6-s + (−1.02 + 1.02i)7-s − 0.382i·8-s + 0.333i·9-s + (−1.49 + 1.49i)10-s + (−0.0424 + 0.0424i)11-s + (0.288 + 0.288i)12-s + 0.245·13-s + (1.34 + 1.34i)14-s + 0.932i·15-s − 1.20·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(867\)    =    \(3 \cdot 17^{2}\)
Sign: $0.943 - 0.332i$
Analytic conductor: \(6.92302\)
Root analytic conductor: \(2.63116\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{867} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 867,\ (\ :1/2),\ 0.943 - 0.332i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.127525 + 0.0217881i\)
\(L(\frac12)\) \(\approx\) \(0.127525 + 0.0217881i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.84iT - 2T^{2} \)
5 \( 1 + (2.55 + 2.55i)T + 5iT^{2} \)
7 \( 1 + (2.72 - 2.72i)T - 7iT^{2} \)
11 \( 1 + (0.140 - 0.140i)T - 11iT^{2} \)
13 \( 1 - 0.883T + 13T^{2} \)
19 \( 1 - 2.24iT - 19T^{2} \)
23 \( 1 + (-1.80 + 1.80i)T - 23iT^{2} \)
29 \( 1 + (-1.58 - 1.58i)T + 29iT^{2} \)
31 \( 1 + (3.35 + 3.35i)T + 31iT^{2} \)
37 \( 1 + (-4.93 - 4.93i)T + 37iT^{2} \)
41 \( 1 + (0.438 - 0.438i)T - 41iT^{2} \)
43 \( 1 + 7.17iT - 43T^{2} \)
47 \( 1 + 5.49T + 47T^{2} \)
53 \( 1 - 5.18iT - 53T^{2} \)
59 \( 1 - 5.01iT - 59T^{2} \)
61 \( 1 + (10.6 - 10.6i)T - 61iT^{2} \)
67 \( 1 + 0.281T + 67T^{2} \)
71 \( 1 + (9.85 + 9.85i)T + 71iT^{2} \)
73 \( 1 + (-11.0 - 11.0i)T + 73iT^{2} \)
79 \( 1 + (-3.62 + 3.62i)T - 79iT^{2} \)
83 \( 1 - 9.51iT - 83T^{2} \)
89 \( 1 + 4.33T + 89T^{2} \)
97 \( 1 + (-4.20 - 4.20i)T + 97iT^{2} \)
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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

−10.37886972820210686318735694707, −9.368673650939279931352508272225, −8.807210948798531548351836866821, −7.85501972348760020637711125214, −6.72528557349076686704663528291, −5.71377505583887423544891198347, −4.59647510367831988283152635474, −3.65343914426058154906818322499, −2.63421034856324729032930441372, −1.22189926938683465050434602663, 0.07248925801187343866692222835, 3.04065906785246803659516070751, 3.86041690952362724884714435460, 4.85767919768233018558446399461, 6.17887829003226040478468366695, 6.70935854214062959581253658506, 7.35064204511555626761227507120, 7.994395496975019558315853512725, 9.190590070277497946351781424637, 10.13780273456760164332206850845

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