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Properties

Label 2-930-155.37-c1-0-29
Degree $2$
Conductor $930$
Sign $0.0846 + 0.996i$
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  + (0.707 + 0.707i)2-s + (−0.258 + 0.965i)3-s + 1.00i·4-s + (0.260 − 2.22i)5-s + (−0.866 + 0.500i)6-s + (0.587 − 2.19i)7-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (1.75 − 1.38i)10-s + (−5.18 − 2.99i)11-s + (−0.965 − 0.258i)12-s + (−0.731 + 0.196i)13-s + (1.96 − 1.13i)14-s + (2.07 + 0.826i)15-s − 1.00·16-s + (−5.99 − 1.60i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.149 + 0.557i)3-s + 0.500i·4-s + (0.116 − 0.993i)5-s + (−0.353 + 0.204i)6-s + (0.222 − 0.828i)7-s + (−0.250 + 0.250i)8-s + (−0.288 − 0.166i)9-s + (0.554 − 0.438i)10-s + (−1.56 − 0.903i)11-s + (−0.278 − 0.0747i)12-s + (−0.202 + 0.0543i)13-s + (0.525 − 0.303i)14-s + (0.536 + 0.213i)15-s − 0.250·16-s + (−1.45 − 0.389i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0846 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0846 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.0846 + 0.996i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.0846 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.752078 - 0.690928i\)
\(L(\frac12)\) \(\approx\) \(0.752078 - 0.690928i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (-0.260 + 2.22i)T \)
31 \( 1 + (5.55 + 0.408i)T \)
good7 \( 1 + (-0.587 + 2.19i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (5.18 + 2.99i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.731 - 0.196i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (5.99 + 1.60i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.07 + 1.19i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.38 + 2.38i)T + 23iT^{2} \)
29 \( 1 + 0.413T + 29T^{2} \)
37 \( 1 + (-3.75 - 1.00i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.22 - 2.12i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.299 - 1.11i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-6.11 - 6.11i)T + 47iT^{2} \)
53 \( 1 + (-0.400 + 0.107i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-9.59 + 5.53i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 1.49iT - 61T^{2} \)
67 \( 1 + (-13.5 + 3.62i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.10 + 1.90i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.84 + 1.56i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (6.77 + 11.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.43 - 0.921i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 + (-7.87 - 7.87i)T + 97iT^{2} \)
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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

−9.821008554156322421048652407329, −8.914760692391443571975444048154, −8.154944316704032131038209713119, −7.41135394006774746270735263066, −6.21922548947736726062476353651, −5.27429803438871733325594936365, −4.72289146883619547234460706868, −3.84002051902259120092047109558, −2.49292753108341977670188092176, −0.37733370718309507873321127740, 2.18547176954869124185916908423, 2.41825614615050756741249256165, 3.85713205495020142114416732808, 5.19909580800350551607380398547, 5.75639384736200489571441870325, 6.87247101321234632196766454334, 7.53715701182267272299671820737, 8.558812930864197469294854063483, 9.701361701208631042834625066344, 10.43840401419183043087436537652

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