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Properties

Label 2-930-93.92-c1-0-8
Degree $2$
Conductor $930$
Sign $-0.323 - 0.946i$
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  − i·2-s + (1.22 + 1.22i)3-s − 4-s + i·5-s + (1.22 − 1.22i)6-s − 4·7-s + i·8-s + 2.99i·9-s + 10-s + 4.89·11-s + (−1.22 − 1.22i)12-s + 2.44i·13-s + 4i·14-s + (−1.22 + 1.22i)15-s + 16-s − 7.34·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.707 + 0.707i)3-s − 0.5·4-s + 0.447i·5-s + (0.499 − 0.499i)6-s − 1.51·7-s + 0.353i·8-s + 0.999i·9-s + 0.316·10-s + 1.47·11-s + (−0.353 − 0.353i)12-s + 0.679i·13-s + 1.06i·14-s + (−0.316 + 0.316i)15-s + 0.250·16-s − 1.78·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.323 - 0.946i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.323 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.591971 + 0.828380i\)
\(L(\frac12)\) \(\approx\) \(0.591971 + 0.828380i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 - iT \)
31 \( 1 + (-5 - 2.44i)T \)
good7 \( 1 + 4T + 7T^{2} \)
11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 + 7.34T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 7.34T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
37 \( 1 + 2.44iT - 37T^{2} \)
41 \( 1 - 12iT - 41T^{2} \)
43 \( 1 + 2.44iT - 43T^{2} \)
47 \( 1 - 12iT - 47T^{2} \)
53 \( 1 + 2.44T + 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 2.44iT - 73T^{2} \)
79 \( 1 - 4.89iT - 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 + 4.89T + 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−10.16950517242938493297769039645, −9.416276840484826405458948905046, −9.141635510260978601417375404278, −8.109859818418778700020152893562, −6.65824643592717563787413846397, −6.28403738718995254044774298390, −4.45656208179113712495322328784, −3.95061852121690728800583033493, −3.01230036477294194359640859200, −2.01208552782541318451489267142, 0.41624285938825664574505994309, 2.16808448627865159263010589620, 3.56935227814330193844270139000, 4.25184918460406237006618727338, 5.93736066453395959914922735478, 6.49812157135989280931032658214, 7.06504132842321946108501167189, 8.244308757897650558632460419559, 8.882160393692731533054478296587, 9.455158530876443027097836969567

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