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Properties

Label 2-930-465.464-c1-0-15
Degree $2$
Conductor $930$
Sign $0.936 + 0.349i$
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 + (−0.230 − 1.71i)3-s + 4-s + (−0.286 − 2.21i)5-s + (0.230 + 1.71i)6-s + 4.09i·7-s − 8-s + (−2.89 + 0.791i)9-s + (0.286 + 2.21i)10-s + 2.91·11-s + (−0.230 − 1.71i)12-s + 2.84·13-s − 4.09i·14-s + (−3.74 + 1.00i)15-s + 16-s + 3.40i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.133 − 0.991i)3-s + 0.5·4-s + (−0.128 − 0.991i)5-s + (0.0941 + 0.700i)6-s + 1.54i·7-s − 0.353·8-s + (−0.964 + 0.263i)9-s + (0.0906 + 0.701i)10-s + 0.879·11-s + (−0.0665 − 0.495i)12-s + 0.790·13-s − 1.09i·14-s + (−0.965 + 0.258i)15-s + 0.250·16-s + 0.826i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.936 + 0.349i$
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: \((2,\ 930,\ (\ :1/2),\ 0.936 + 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06886 - 0.193114i\)
\(L(\frac12)\) \(\approx\) \(1.06886 - 0.193114i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.230 + 1.71i)T \)
5 \( 1 + (0.286 + 2.21i)T \)
31 \( 1 + (-4.53 - 3.23i)T \)
good7 \( 1 - 4.09iT - 7T^{2} \)
11 \( 1 - 2.91T + 11T^{2} \)
13 \( 1 - 2.84T + 13T^{2} \)
17 \( 1 - 3.40iT - 17T^{2} \)
19 \( 1 - 2.22T + 19T^{2} \)
23 \( 1 - 3.74iT - 23T^{2} \)
29 \( 1 - 5.67T + 29T^{2} \)
37 \( 1 + 1.51T + 37T^{2} \)
41 \( 1 - 0.438iT - 41T^{2} \)
43 \( 1 - 8.29T + 43T^{2} \)
47 \( 1 + 7.81T + 47T^{2} \)
53 \( 1 + 10.9iT - 53T^{2} \)
59 \( 1 - 2.26iT - 59T^{2} \)
61 \( 1 - 3.00iT - 61T^{2} \)
67 \( 1 + 9.68iT - 67T^{2} \)
71 \( 1 + 11.2iT - 71T^{2} \)
73 \( 1 - 2.62T + 73T^{2} \)
79 \( 1 - 15.1iT - 79T^{2} \)
83 \( 1 + 8.73iT - 83T^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + 3.32iT - 97T^{2} \)
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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

−9.702445423994641554230353509060, −8.867286945734831890723274347566, −8.514083765202961143746998606522, −7.77960589614852968898705224200, −6.48875969429138337877771935389, −5.96539437140837434313816210474, −5.03620541847324555311700994594, −3.37690208227471016501514541384, −2.00815502940552715508706285633, −1.12265851557532272583680518858, 0.836098398904160244163182128834, 2.84331097644270054226744115771, 3.76432759935484559013694968696, 4.53577470255536523469572645871, 6.09299411014723615652975843832, 6.76006348745639213390844909942, 7.58777095280475839778139446740, 8.546386558455283122438405455675, 9.544626280579902772155868959258, 10.14399356718000379148829122810

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