Properties

Label 2-930-5.4-c1-0-5
Degree $2$
Conductor $930$
Sign $-0.447 - 0.894i$
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 + (1 + 2i)5-s + 6-s + 5i·7-s + i·8-s − 9-s + (2 − i)10-s − 11-s i·12-s + 5·14-s + (−2 + i)15-s + 16-s − 4i·17-s + i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.447 + 0.894i)5-s + 0.408·6-s + 1.88i·7-s + 0.353i·8-s − 0.333·9-s + (0.632 − 0.316i)10-s − 0.301·11-s − 0.288i·12-s + 1.33·14-s + (−0.516 + 0.258i)15-s + 0.250·16-s − 0.970i·17-s + 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.447 - 0.894i$
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: \((2,\ 930,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.603105 + 0.975845i\)
\(L(\frac12)\) \(\approx\) \(0.603105 + 0.975845i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 2i)T \)
31 \( 1 + T \)
good7 \( 1 - 5iT - 7T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 + 3T + 19T^{2} \)
23 \( 1 - iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 + 3iT - 53T^{2} \)
59 \( 1 - 14T + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 - 10iT - 67T^{2} \)
71 \( 1 - 9T + 71T^{2} \)
73 \( 1 - 7iT - 73T^{2} \)
79 \( 1 + 15T + 79T^{2} \)
83 \( 1 - 10iT - 83T^{2} \)
89 \( 1 - T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−10.28333496243343160019385547601, −9.533836145126046391854071597588, −9.027610487362500617128024115490, −8.106490942136047456984712173931, −6.81544539371626149443485786711, −5.65985739249986372350084822645, −5.27927439793987147849687201147, −3.82969445389007630682392656451, −2.69758095415948915636767956929, −2.21241908536177274458602143941, 0.52648655962142875337134174249, 1.76117529757177652857111550953, 3.73543790169156736936307645155, 4.50740365625074095957041183920, 5.55733500803930196949177533093, 6.49069888343439444986924777984, 7.24458161526450368608660492853, 8.040791268105960274438525677327, 8.671093947431050969266309820508, 9.811493788425270931391302038918

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