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Properties

Label 2-931-133.132-c1-0-39
Degree $2$
Conductor $931$
Sign $0.174 + 0.984i$
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  + 0.196i·2-s − 2.78·3-s + 1.96·4-s − 2.75i·5-s − 0.548i·6-s + 0.778i·8-s + 4.78·9-s + 0.540·10-s − 2.07·11-s − 5.47·12-s + 5.35·13-s + 7.67i·15-s + 3.76·16-s − 3.70i·17-s + 0.939i·18-s + (−1.06 + 4.22i)19-s + ⋯
L(s)  = 1  + 0.138i·2-s − 1.61·3-s + 0.980·4-s − 1.22i·5-s − 0.223i·6-s + 0.275i·8-s + 1.59·9-s + 0.170·10-s − 0.626·11-s − 1.57·12-s + 1.48·13-s + 1.98i·15-s + 0.942·16-s − 0.899i·17-s + 0.221i·18-s + (−0.243 + 0.969i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.174 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $0.174 + 0.984i$
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: \((2,\ 931,\ (\ :1/2),\ 0.174 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.818481 - 0.686396i\)
\(L(\frac12)\) \(\approx\) \(0.818481 - 0.686396i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.06 - 4.22i)T \)
good2 \( 1 - 0.196iT - 2T^{2} \)
3 \( 1 + 2.78T + 3T^{2} \)
5 \( 1 + 2.75iT - 5T^{2} \)
11 \( 1 + 2.07T + 11T^{2} \)
13 \( 1 - 5.35T + 13T^{2} \)
17 \( 1 + 3.70iT - 17T^{2} \)
23 \( 1 - 1.06T + 23T^{2} \)
29 \( 1 + 7.59iT - 29T^{2} \)
31 \( 1 + 9.90T + 31T^{2} \)
37 \( 1 + 2.85iT - 37T^{2} \)
41 \( 1 - 4.10T + 41T^{2} \)
43 \( 1 + 2.06T + 43T^{2} \)
47 \( 1 + 8.20iT - 47T^{2} \)
53 \( 1 + 7.36iT - 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 - 11.2iT - 61T^{2} \)
67 \( 1 + 8.91iT - 67T^{2} \)
71 \( 1 + 3.64iT - 71T^{2} \)
73 \( 1 + 12.9iT - 73T^{2} \)
79 \( 1 + 13.1iT - 79T^{2} \)
83 \( 1 + 10.4iT - 83T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 + 1.90T + 97T^{2} \)
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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

−10.22170646287047176733629851576, −9.044378465706906766209757742992, −8.059911143236447536157418550955, −7.18766867566794742965470652893, −6.13991564992061089701413640543, −5.66819536512997344353584032439, −4.93811294939489502129102529965, −3.73209936809147928652620455052, −1.84952085173172726870682661099, −0.65912361672142660015890005371, 1.37315420716796140952154382023, 2.81813498133760275322286303195, 3.89483410713746001425352323935, 5.39185962965878668783547098291, 6.02354649260038431747305395890, 6.78799222139467814428796502428, 7.18796521851750524364087549214, 8.470828578177463071795633492232, 9.929706467051625196410258362731, 10.83605811055500124514949135620

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