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Properties

Label 2-931-133.132-c1-0-58
Degree $2$
Conductor $931$
Sign $-0.458 - 0.888i$
Analytic cond. $7.43407$
Root an. cond. $2.72654$
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  − 1.38i·2-s − 2.17·3-s + 0.0839·4-s − 4.26i·5-s + 3.01i·6-s − 2.88i·8-s + 1.74·9-s − 5.90·10-s + 0.143·11-s − 0.182·12-s − 2.01·13-s + 9.30i·15-s − 3.82·16-s − 2.60i·17-s − 2.42i·18-s + (−3.40 − 2.71i)19-s + ⋯
L(s)  = 1  − 0.978i·2-s − 1.25·3-s + 0.0419·4-s − 1.90i·5-s + 1.23i·6-s − 1.01i·8-s + 0.582·9-s − 1.86·10-s + 0.0431·11-s − 0.0528·12-s − 0.559·13-s + 2.40i·15-s − 0.956·16-s − 0.631i·17-s − 0.570i·18-s + (−0.782 − 0.623i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 - 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $-0.458 - 0.888i$
Analytic conductor: \(7.43407\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (930, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :1/2),\ -0.458 - 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.396065 + 0.650282i\)
\(L(\frac12)\) \(\approx\) \(0.396065 + 0.650282i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 + (3.40 + 2.71i)T \)
good2 \( 1 + 1.38iT - 2T^{2} \)
3 \( 1 + 2.17T + 3T^{2} \)
5 \( 1 + 4.26iT - 5T^{2} \)
11 \( 1 - 0.143T + 11T^{2} \)
13 \( 1 + 2.01T + 13T^{2} \)
17 \( 1 + 2.60iT - 17T^{2} \)
23 \( 1 - 7.11T + 23T^{2} \)
29 \( 1 + 7.53iT - 29T^{2} \)
31 \( 1 - 5.35T + 31T^{2} \)
37 \( 1 - 9.99iT - 37T^{2} \)
41 \( 1 + 1.27T + 41T^{2} \)
43 \( 1 - 1.66T + 43T^{2} \)
47 \( 1 - 7.99iT - 47T^{2} \)
53 \( 1 - 5.72iT - 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 - 6.32iT - 61T^{2} \)
67 \( 1 + 5.56iT - 67T^{2} \)
71 \( 1 + 4.99iT - 71T^{2} \)
73 \( 1 - 3.74iT - 73T^{2} \)
79 \( 1 - 1.47iT - 79T^{2} \)
83 \( 1 - 2.80iT - 83T^{2} \)
89 \( 1 - 2.80T + 89T^{2} \)
97 \( 1 - 11.9T + 97T^{2} \)
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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

−9.614283704846005201676965229560, −9.015020054925172655405646266649, −7.923629411237809213519009454165, −6.72762605109649725444594645062, −5.86574515683468142785168447890, −4.72543852376907915257222710166, −4.53731520365921241913525005318, −2.75883973819364428770038893370, −1.29539309276089863021878423773, −0.44125663999454646914150072141, 2.21283395657664593672595915093, 3.38104779101930789726865347326, 4.89681571275827904364757776118, 5.82865481829573021424016366667, 6.42873744046200093472105440772, 6.97631662380131353685102532760, 7.60578629561899230957794762186, 8.769590134612348665117329158747, 10.22387768505683550481142342729, 10.70726226888813576643571564495

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