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Properties

Label 2-930-5.4-c1-0-20
Degree $2$
Conductor $930$
Sign $-0.270 + 0.962i$
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 + (0.605 − 2.15i)5-s + 6-s − 2i·7-s + i·8-s − 9-s + (−2.15 − 0.605i)10-s + 5.21·11-s − i·12-s + 0.789i·13-s − 2·14-s + (2.15 + 0.605i)15-s + 16-s + 0.115i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.270 − 0.962i)5-s + 0.408·6-s − 0.755i·7-s + 0.353i·8-s − 0.333·9-s + (−0.680 − 0.191i)10-s + 1.57·11-s − 0.288i·12-s + 0.219i·13-s − 0.534·14-s + (0.555 + 0.156i)15-s + 0.250·16-s + 0.0279i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.270 + 0.962i$
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: \((2,\ 930,\ (\ :1/2),\ -0.270 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.929058 - 1.22621i\)
\(L(\frac12)\) \(\approx\) \(0.929058 - 1.22621i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.605 + 2.15i)T \)
31 \( 1 + T \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 5.21T + 11T^{2} \)
13 \( 1 - 0.789iT - 13T^{2} \)
17 \( 1 - 0.115iT - 17T^{2} \)
19 \( 1 + 0.115T + 19T^{2} \)
23 \( 1 + 4.42iT - 23T^{2} \)
29 \( 1 + 4.42T + 29T^{2} \)
37 \( 1 + 6.61iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 8.61iT - 43T^{2} \)
47 \( 1 + 0.115iT - 47T^{2} \)
53 \( 1 + 0.190iT - 53T^{2} \)
59 \( 1 - 4.19T + 59T^{2} \)
61 \( 1 - 12.4T + 61T^{2} \)
67 \( 1 + 5.82iT - 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + 10.8iT - 73T^{2} \)
79 \( 1 + 2.30T + 79T^{2} \)
83 \( 1 - 15.1iT - 83T^{2} \)
89 \( 1 - 2.23T + 89T^{2} \)
97 \( 1 - 9.21iT - 97T^{2} \)
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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

−9.809777742966365924996867433581, −9.078755497866409856467530079804, −8.627068987869605110186483465195, −7.35461894735942099701605310397, −6.22156536663110368332045121950, −5.17898751134358155285796112113, −4.15923503549313541264775914404, −3.77988037861326749413454058666, −2.04695849668419524796122432955, −0.78776609613215060268633625533, 1.57738892623066139293423674829, 2.94188179586906768208890435097, 4.01845292749549910138937693410, 5.49569378576851641124442416441, 6.14729178059506032022686174371, 6.86196237297376117471208480274, 7.56655818534354642520834219571, 8.601683032687253795659550243411, 9.338121325944569389627327126008, 10.08306726759127149652436084652

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