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Properties

Label 2-931-133.132-c1-0-48
Degree $2$
Conductor $931$
Sign $0.362 + 0.932i$
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  − 0.777i·2-s + 1.65·3-s + 1.39·4-s − 0.862i·5-s − 1.28i·6-s − 2.63i·8-s − 0.247·9-s − 0.670·10-s + 3.26·11-s + 2.31·12-s − 1.52·13-s − 1.43i·15-s + 0.739·16-s + 1.16i·17-s + 0.192i·18-s + (3.76 − 2.19i)19-s + ⋯
L(s)  = 1  − 0.549i·2-s + 0.957·3-s + 0.697·4-s − 0.385i·5-s − 0.526i·6-s − 0.933i·8-s − 0.0824·9-s − 0.212·10-s + 0.985·11-s + 0.668·12-s − 0.421·13-s − 0.369i·15-s + 0.184·16-s + 0.283i·17-s + 0.0453i·18-s + (0.863 − 0.503i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.362 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.362 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $0.362 + 0.932i$
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: \((2,\ 931,\ (\ :1/2),\ 0.362 + 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18930 - 1.49789i\)
\(L(\frac12)\) \(\approx\) \(2.18930 - 1.49789i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.76 + 2.19i)T \)
good2 \( 1 + 0.777iT - 2T^{2} \)
3 \( 1 - 1.65T + 3T^{2} \)
5 \( 1 + 0.862iT - 5T^{2} \)
11 \( 1 - 3.26T + 11T^{2} \)
13 \( 1 + 1.52T + 13T^{2} \)
17 \( 1 - 1.16iT - 17T^{2} \)
23 \( 1 + 0.656T + 23T^{2} \)
29 \( 1 + 3.23iT - 29T^{2} \)
31 \( 1 + 3.45T + 31T^{2} \)
37 \( 1 - 2.41iT - 37T^{2} \)
41 \( 1 + 8.18T + 41T^{2} \)
43 \( 1 - 7.79T + 43T^{2} \)
47 \( 1 - 5.03iT - 47T^{2} \)
53 \( 1 - 1.56iT - 53T^{2} \)
59 \( 1 + 0.438T + 59T^{2} \)
61 \( 1 - 8.06iT - 61T^{2} \)
67 \( 1 + 8.73iT - 67T^{2} \)
71 \( 1 - 2.50iT - 71T^{2} \)
73 \( 1 - 7.67iT - 73T^{2} \)
79 \( 1 - 13.8iT - 79T^{2} \)
83 \( 1 + 0.260iT - 83T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 + 3.19T + 97T^{2} \)
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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

−9.726608365144054877975441837302, −9.221795027957535388209598085422, −8.349106696835593618947154352254, −7.43501784292491240518237621646, −6.64805114174564560902582976968, −5.55405676352755608367490679261, −4.20700977369158025900603191635, −3.27223993896320958289454724774, −2.43264443575825872480668328540, −1.25306012979062017340849348233, 1.76423152165334989162938044905, 2.86472492875270569207302822889, 3.63612307449929215486522862509, 5.13584229470078393086718975031, 6.09468688695070521387631385634, 7.03592976673598899013218487244, 7.54735351616602170402082131547, 8.511081213884584449810420591064, 9.168801374647177971027685935930, 10.10738177590611739120801130097

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