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Properties

Label 2-931-7.4-c1-0-14
Degree $2$
Conductor $931$
Sign $0.266 + 0.963i$
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  + (1.30 + 2.26i)2-s + (−1.30 + 2.26i)3-s + (−2.42 + 4.20i)4-s + (1.11 + 1.93i)5-s − 6.85·6-s − 7.47·8-s + (−1.92 − 3.33i)9-s + (−2.92 + 5.06i)10-s + (2.80 − 4.86i)11-s + (−6.35 − 11.0i)12-s − 13-s − 5.85·15-s + (−4.92 − 8.53i)16-s + (−2.42 + 4.20i)17-s + (5.04 − 8.73i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (0.925 + 1.60i)2-s + (−0.755 + 1.30i)3-s + (−1.21 + 2.10i)4-s + (0.499 + 0.866i)5-s − 2.79·6-s − 2.64·8-s + (−0.642 − 1.11i)9-s + (−0.925 + 1.60i)10-s + (0.846 − 1.46i)11-s + (−1.83 − 3.17i)12-s − 0.277·13-s − 1.51·15-s + (−1.23 − 2.13i)16-s + (−0.588 + 1.01i)17-s + (1.18 − 2.05i)18-s + (−0.114 − 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $0.266 + 0.963i$
Analytic conductor: \(7.43407\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (704, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :1/2),\ 0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30671 - 0.994089i\)
\(L(\frac12)\) \(\approx\) \(1.30671 - 0.994089i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-1.30 - 2.26i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.30 - 2.26i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.11 - 1.93i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.80 + 4.86i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (2.42 - 4.20i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.61T + 29T^{2} \)
31 \( 1 + (0.427 - 0.739i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.35 - 9.27i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.381T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + (-5.59 - 9.68i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.545 - 0.944i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.354 + 0.613i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.35 + 5.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.28 - 5.68i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.47T + 71T^{2} \)
73 \( 1 + (-2.07 + 3.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.85T + 83T^{2} \)
89 \( 1 + (-1.14 - 1.98i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14.4T + 97T^{2} \)
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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

−10.90718146733355147566828814260, −9.743869076087980781946571888448, −8.974580631859475848333286936210, −8.085993840725804947400037256986, −6.85999201302172743036292875615, −6.14217498086037559948020517768, −5.78037647086955773906929587104, −4.75209375379483682850686890840, −3.92128130879223778504777380008, −3.15714738709712347606434567235, 0.63564256429663016229535194795, 1.73210041136115963931350398325, 2.30724127734697957801375055040, 4.03581133418457785877301080630, 4.92038500154101764890966539901, 5.56900259590831948666934771216, 6.60190952824381732880754182278, 7.44899061934584344894117090404, 9.128423253861746337120930793921, 9.471545840926141479630714793074

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