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Properties

Label 2-930-465.464-c1-0-10
Degree $2$
Conductor $930$
Sign $-0.825 - 0.563i$
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 + (0.880 + 1.49i)3-s + 4-s + (−2.09 + 0.775i)5-s + (0.880 + 1.49i)6-s + 0.657i·7-s + 8-s + (−1.44 + 2.62i)9-s + (−2.09 + 0.775i)10-s − 3.48·11-s + (0.880 + 1.49i)12-s − 4.74·13-s + 0.657i·14-s + (−3.00 − 2.44i)15-s + 16-s + 3.35i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.508 + 0.861i)3-s + 0.5·4-s + (−0.937 + 0.346i)5-s + (0.359 + 0.608i)6-s + 0.248i·7-s + 0.353·8-s + (−0.482 + 0.875i)9-s + (−0.663 + 0.245i)10-s − 1.05·11-s + (0.254 + 0.430i)12-s − 1.31·13-s + 0.175i·14-s + (−0.775 − 0.631i)15-s + 0.250·16-s + 0.814i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.825 - 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.825 - 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.825 - 0.563i$
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: \((2,\ 930,\ (\ :1/2),\ -0.825 - 0.563i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.488113 + 1.58111i\)
\(L(\frac12)\) \(\approx\) \(0.488113 + 1.58111i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.880 - 1.49i)T \)
5 \( 1 + (2.09 - 0.775i)T \)
31 \( 1 + (-5.54 + 0.469i)T \)
good7 \( 1 - 0.657iT - 7T^{2} \)
11 \( 1 + 3.48T + 11T^{2} \)
13 \( 1 + 4.74T + 13T^{2} \)
17 \( 1 - 3.35iT - 17T^{2} \)
19 \( 1 - 0.313T + 19T^{2} \)
23 \( 1 - 6.17iT - 23T^{2} \)
29 \( 1 - 1.48T + 29T^{2} \)
37 \( 1 - 4.95T + 37T^{2} \)
41 \( 1 + 0.355iT - 41T^{2} \)
43 \( 1 + 4.78T + 43T^{2} \)
47 \( 1 - 0.472T + 47T^{2} \)
53 \( 1 - 3.48iT - 53T^{2} \)
59 \( 1 + 6.24iT - 59T^{2} \)
61 \( 1 - 3.55iT - 61T^{2} \)
67 \( 1 - 15.3iT - 67T^{2} \)
71 \( 1 - 7.71iT - 71T^{2} \)
73 \( 1 + 3.30T + 73T^{2} \)
79 \( 1 + 13.9iT - 79T^{2} \)
83 \( 1 - 7.45iT - 83T^{2} \)
89 \( 1 - 0.308T + 89T^{2} \)
97 \( 1 - 10.7iT - 97T^{2} \)
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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

−10.35509225537626123500692890969, −9.843935654453652268322688474225, −8.587780125165600268083911158776, −7.83142643241362604564925139581, −7.20750944829280267972630101470, −5.80635280822921877715668429588, −4.92856953892049019524677569574, −4.18210916783685824678955075897, −3.16454010109365939640707592362, −2.41953434943143721497413544450, 0.54291332228322121570497180233, 2.39229711411564261493828440381, 3.11960460033156185139166111585, 4.44240456759189290594385827314, 5.12832278104460511181121818180, 6.45102033958233800421405323609, 7.30934041475242568445911701678, 7.80988262625027138181077075989, 8.594354519270694979432627178237, 9.735132464640971231615897164268

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