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Properties

Label 2-930-5.4-c1-0-8
Degree $2$
Conductor $930$
Sign $0.999 + 0.0235i$
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 + i·3-s − 4-s + (−2.23 − 0.0526i)5-s + 6-s − 2i·7-s + i·8-s − 9-s + (−0.0526 + 2.23i)10-s − 0.470·11-s − i·12-s + 6.47i·13-s − 2·14-s + (0.0526 − 2.23i)15-s + 16-s − 7.04i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.999 − 0.0235i)5-s + 0.408·6-s − 0.755i·7-s + 0.353i·8-s − 0.333·9-s + (−0.0166 + 0.706i)10-s − 0.141·11-s − 0.288i·12-s + 1.79i·13-s − 0.534·14-s + (0.0135 − 0.577i)15-s + 0.250·16-s − 1.70i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.999 + 0.0235i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.999 + 0.0235i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15891 - 0.0136488i\)
\(L(\frac12)\) \(\approx\) \(1.15891 - 0.0136488i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 + (2.23 + 0.0526i)T \)
31 \( 1 + T \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 0.470T + 11T^{2} \)
13 \( 1 - 6.47iT - 13T^{2} \)
17 \( 1 + 7.04iT - 17T^{2} \)
19 \( 1 - 7.04T + 19T^{2} \)
23 \( 1 - 6.94iT - 23T^{2} \)
29 \( 1 - 6.94T + 29T^{2} \)
37 \( 1 - 1.78iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 0.210iT - 43T^{2} \)
47 \( 1 - 7.04iT - 47T^{2} \)
53 \( 1 + 3.15iT - 53T^{2} \)
59 \( 1 - 7.15T + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 - 8.26iT - 67T^{2} \)
71 \( 1 - 0.260T + 71T^{2} \)
73 \( 1 - 11.8iT - 73T^{2} \)
79 \( 1 - 1.89T + 79T^{2} \)
83 \( 1 + 11.7iT - 83T^{2} \)
89 \( 1 + 12.0T + 89T^{2} \)
97 \( 1 - 3.52iT - 97T^{2} \)
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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

−9.932713248122373518705156459661, −9.489031096826429258656131482698, −8.616980038011703230432543684148, −7.46318202480501323994333836035, −6.97300332411234527161313936242, −5.28579331807154264471005186648, −4.50416628739632516664040818881, −3.76122966367400274574582673828, −2.82763970041976481777680247682, −1.01461484730051513890731435765, 0.75028415678710255794022844439, 2.73154872617466493573894720637, 3.72840688322061401172338255051, 5.05774721293566686167443644511, 5.79443270245837435101869813423, 6.71097129984658726387873078418, 7.70603399963008599449560404552, 8.230007075327193991581821151282, 8.749850019418616354169742231637, 10.12461864792402564012301247190

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