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Properties

Label 2-930-465.464-c1-0-5
Degree $2$
Conductor $930$
Sign $0.643 - 0.765i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
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  − 2-s + (−1.69 − 0.341i)3-s + 4-s + (−1.23 − 1.86i)5-s + (1.69 + 0.341i)6-s + 0.843i·7-s − 8-s + (2.76 + 1.16i)9-s + (1.23 + 1.86i)10-s − 2.73·11-s + (−1.69 − 0.341i)12-s − 1.61·13-s − 0.843i·14-s + (1.46 + 3.58i)15-s + 16-s − 3.82i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.980 − 0.197i)3-s + 0.5·4-s + (−0.554 − 0.832i)5-s + (0.693 + 0.139i)6-s + 0.318i·7-s − 0.353·8-s + (0.922 + 0.387i)9-s + (0.392 + 0.588i)10-s − 0.824·11-s + (−0.490 − 0.0987i)12-s − 0.448·13-s − 0.225i·14-s + (0.379 + 0.925i)15-s + 0.250·16-s − 0.927i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.643 - 0.765i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.643 - 0.765i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.371782 + 0.173133i\)
\(L(\frac12)\) \(\approx\) \(0.371782 + 0.173133i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (1.69 + 0.341i)T \)
5 \( 1 + (1.23 + 1.86i)T \)
31 \( 1 + (-2.58 - 4.93i)T \)
good7 \( 1 - 0.843iT - 7T^{2} \)
11 \( 1 + 2.73T + 11T^{2} \)
13 \( 1 + 1.61T + 13T^{2} \)
17 \( 1 + 3.82iT - 17T^{2} \)
19 \( 1 + 0.779T + 19T^{2} \)
23 \( 1 - 3.42iT - 23T^{2} \)
29 \( 1 + 4.68T + 29T^{2} \)
37 \( 1 + 4.77T + 37T^{2} \)
41 \( 1 + 7.71iT - 41T^{2} \)
43 \( 1 - 6.37T + 43T^{2} \)
47 \( 1 - 12.0T + 47T^{2} \)
53 \( 1 + 1.15iT - 53T^{2} \)
59 \( 1 - 3.35iT - 59T^{2} \)
61 \( 1 - 15.3iT - 61T^{2} \)
67 \( 1 - 8.81iT - 67T^{2} \)
71 \( 1 - 13.6iT - 71T^{2} \)
73 \( 1 - 11.0T + 73T^{2} \)
79 \( 1 + 4.05iT - 79T^{2} \)
83 \( 1 - 8.96iT - 83T^{2} \)
89 \( 1 - 4.16T + 89T^{2} \)
97 \( 1 + 0.560iT - 97T^{2} \)
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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

−10.25583486181412652633388950818, −9.309135352022141946720353320304, −8.577468847440817935312519729911, −7.45486364213016769897400584062, −7.16883030536973469916997752865, −5.64788161865931588905015070283, −5.23714043443732570498183768740, −4.04922024409696016877238016170, −2.39439713963163828070030164714, −0.920135161552059645037804264921, 0.36420461387312381893845978819, 2.21990252614301362347846615458, 3.60724719900510822502990933446, 4.63222685754862122515418806440, 5.86371730691939465978880931968, 6.58841572545136166011360842614, 7.48206349708135243618945236928, 8.034068068385888854654164322621, 9.324494253076679379381052834019, 10.23998476331590821600106552886

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