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Properties

Label 2-930-5.4-c3-0-66
Degree $2$
Conductor $930$
Sign $0.323 + 0.946i$
Analytic cond. $54.8717$
Root an. cond. $7.40754$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s + 3i·3-s − 4·4-s + (3.61 + 10.5i)5-s + 6·6-s + 2.56i·7-s + 8i·8-s − 9·9-s + (21.1 − 7.22i)10-s + 31.3·11-s − 12i·12-s − 63.6i·13-s + 5.13·14-s + (−31.7 + 10.8i)15-s + 16·16-s + 2.00i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.323 + 0.946i)5-s + 0.408·6-s + 0.138i·7-s + 0.353i·8-s − 0.333·9-s + (0.669 − 0.228i)10-s + 0.859·11-s − 0.288i·12-s − 1.35i·13-s + 0.0979·14-s + (−0.546 + 0.186i)15-s + 0.250·16-s + 0.0285i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.323 + 0.946i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.323 + 0.946i$
Analytic conductor: \(54.8717\)
Root analytic conductor: \(7.40754\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :3/2),\ 0.323 + 0.946i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.651915309\)
\(L(\frac12)\) \(\approx\) \(1.651915309\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
3 \( 1 - 3iT \)
5 \( 1 + (-3.61 - 10.5i)T \)
31 \( 1 + 31T \)
good7 \( 1 - 2.56iT - 343T^{2} \)
11 \( 1 - 31.3T + 1.33e3T^{2} \)
13 \( 1 + 63.6iT - 2.19e3T^{2} \)
17 \( 1 - 2.00iT - 4.91e3T^{2} \)
19 \( 1 + 120.T + 6.85e3T^{2} \)
23 \( 1 + 218. iT - 1.21e4T^{2} \)
29 \( 1 - 200.T + 2.43e4T^{2} \)
37 \( 1 + 90.6iT - 5.06e4T^{2} \)
41 \( 1 + 213.T + 6.89e4T^{2} \)
43 \( 1 - 422. iT - 7.95e4T^{2} \)
47 \( 1 + 264. iT - 1.03e5T^{2} \)
53 \( 1 + 583. iT - 1.48e5T^{2} \)
59 \( 1 - 449.T + 2.05e5T^{2} \)
61 \( 1 - 10.9T + 2.26e5T^{2} \)
67 \( 1 + 656. iT - 3.00e5T^{2} \)
71 \( 1 + 533.T + 3.57e5T^{2} \)
73 \( 1 - 508. iT - 3.89e5T^{2} \)
79 \( 1 - 1.13e3T + 4.93e5T^{2} \)
83 \( 1 + 110. iT - 5.71e5T^{2} \)
89 \( 1 - 1.00e3T + 7.04e5T^{2} \)
97 \( 1 + 1.22e3iT - 9.12e5T^{2} \)
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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

−9.852673994987570417270447707558, −8.754652649182303918665956096534, −8.176722751319210981362644486767, −6.73668700318546205692411310482, −6.09532832052896860130585913094, −4.92263764769484006789357086021, −3.94775200104435884269249858135, −3.00485020042966889122157913113, −2.16479731392275162047525133518, −0.47842539147599849016956979116, 1.06397607911598969809347697106, 1.99695449108431970015724410701, 3.84232309879441524148302393307, 4.62130881889891151933080910376, 5.69092232485707462439511889998, 6.48988383210393292674897110567, 7.17098230172934103072956153703, 8.222729430695787399446704990394, 8.957503950721843841289720695035, 9.416444077138150230112351819633

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