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Properties

Label 2-930-15.2-c1-0-14
Degree $2$
Conductor $930$
Sign $-0.896 - 0.442i$
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  + (−0.707 + 0.707i)2-s + (0.765 + 1.55i)3-s − 1.00i·4-s + (1.73 + 1.41i)5-s + (−1.63 − 0.557i)6-s + (1.07 + 1.07i)7-s + (0.707 + 0.707i)8-s + (−1.82 + 2.37i)9-s + (−2.22 + 0.226i)10-s − 5.88i·11-s + (1.55 − 0.765i)12-s + (−2.66 + 2.66i)13-s − 1.52·14-s + (−0.870 + 3.77i)15-s − 1.00·16-s + (−4.04 + 4.04i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.441 + 0.897i)3-s − 0.500i·4-s + (0.775 + 0.631i)5-s + (−0.669 − 0.227i)6-s + (0.407 + 0.407i)7-s + (0.250 + 0.250i)8-s + (−0.609 + 0.792i)9-s + (−0.703 + 0.0715i)10-s − 1.77i·11-s + (0.448 − 0.220i)12-s + (−0.738 + 0.738i)13-s − 0.407·14-s + (−0.224 + 0.974i)15-s − 0.250·16-s + (−0.980 + 0.980i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.896 - 0.442i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.896 - 0.442i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.319417 + 1.36912i\)
\(L(\frac12)\) \(\approx\) \(0.319417 + 1.36912i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.765 - 1.55i)T \)
5 \( 1 + (-1.73 - 1.41i)T \)
31 \( 1 - T \)
good7 \( 1 + (-1.07 - 1.07i)T + 7iT^{2} \)
11 \( 1 + 5.88iT - 11T^{2} \)
13 \( 1 + (2.66 - 2.66i)T - 13iT^{2} \)
17 \( 1 + (4.04 - 4.04i)T - 17iT^{2} \)
19 \( 1 - 3.95iT - 19T^{2} \)
23 \( 1 + (-1.71 - 1.71i)T + 23iT^{2} \)
29 \( 1 + 1.71T + 29T^{2} \)
37 \( 1 + (-6.38 - 6.38i)T + 37iT^{2} \)
41 \( 1 + 2.84iT - 41T^{2} \)
43 \( 1 + (0.535 - 0.535i)T - 43iT^{2} \)
47 \( 1 + (1.30 - 1.30i)T - 47iT^{2} \)
53 \( 1 + (-1.86 - 1.86i)T + 53iT^{2} \)
59 \( 1 - 6.57T + 59T^{2} \)
61 \( 1 - 3.80T + 61T^{2} \)
67 \( 1 + (-8.88 - 8.88i)T + 67iT^{2} \)
71 \( 1 + 5.12iT - 71T^{2} \)
73 \( 1 + (5.41 - 5.41i)T - 73iT^{2} \)
79 \( 1 - 16.1iT - 79T^{2} \)
83 \( 1 + (11.7 + 11.7i)T + 83iT^{2} \)
89 \( 1 - 1.16T + 89T^{2} \)
97 \( 1 + (9.95 + 9.95i)T + 97iT^{2} \)
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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

−10.16893744814847933352011939638, −9.608785451907266730585564089212, −8.610061540476774140831894257385, −8.343514072575418819400917706277, −7.02878736958075609974907542190, −5.99007007122959014259840502498, −5.47676025226221553049336332831, −4.21638080608658306373544105395, −3.01329117932539267909622514741, −1.91293079269456395160470063816, 0.71929099204203074618735655311, 2.05454099677814943371873979540, 2.61055116198133797788557956352, 4.39593570972551806492663505866, 5.15780434105355953626011399669, 6.65166152871177858504385313936, 7.30270994262629115378194125695, 8.010647149099830042412097132967, 9.126670240208692460248510369445, 9.474829397546112026461020026075

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