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Properties

Label 2-930-31.2-c1-0-1
Degree $2$
Conductor $930$
Sign $-0.536 - 0.843i$
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  + (−0.809 − 0.587i)2-s + (−0.809 + 0.587i)3-s + (0.309 + 0.951i)4-s + 5-s + 6-s + (1.03 + 3.18i)7-s + (0.309 − 0.951i)8-s + (0.309 − 0.951i)9-s + (−0.809 − 0.587i)10-s + (1.50 + 4.62i)11-s + (−0.809 − 0.587i)12-s + (−2.46 + 1.79i)13-s + (1.03 − 3.18i)14-s + (−0.809 + 0.587i)15-s + (−0.809 + 0.587i)16-s + (−1.99 + 6.14i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.467 + 0.339i)3-s + (0.154 + 0.475i)4-s + 0.447·5-s + 0.408·6-s + (0.391 + 1.20i)7-s + (0.109 − 0.336i)8-s + (0.103 − 0.317i)9-s + (−0.255 − 0.185i)10-s + (0.453 + 1.39i)11-s + (−0.233 − 0.169i)12-s + (−0.684 + 0.497i)13-s + (0.276 − 0.850i)14-s + (−0.208 + 0.151i)15-s + (−0.202 + 0.146i)16-s + (−0.484 + 1.49i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.536 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.536 - 0.843i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.536 - 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.365930 + 0.666350i\)
\(L(\frac12)\) \(\approx\) \(0.365930 + 0.666350i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 - T \)
31 \( 1 + (-0.262 + 5.56i)T \)
good7 \( 1 + (-1.03 - 3.18i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-1.50 - 4.62i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (2.46 - 1.79i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.99 - 6.14i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (6.48 + 4.71i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-2.45 + 7.54i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-3.26 - 2.37i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + 4.97T + 37T^{2} \)
41 \( 1 + (-0.430 - 0.312i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-1.69 - 1.23i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (10.9 - 7.95i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.98 - 6.11i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-2.86 + 2.08i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + 12.8T + 61T^{2} \)
67 \( 1 + 1.70T + 67T^{2} \)
71 \( 1 + (-2.29 + 7.07i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.21 - 9.90i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1.45 + 4.48i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-6.03 - 4.38i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-2.48 - 7.63i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-0.438 - 1.34i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.45007213285692265041997776374, −9.400789098742120648281473838039, −8.976912834451037018533195891939, −8.119487898665125426675817725105, −6.72505686963419506783736786550, −6.30261099198249739961413682898, −4.83691363607490912419269925480, −4.35704374785865385884021966989, −2.51374201221463523715342545465, −1.85014399333912889905844877017, 0.44973073691972461788629958467, 1.66497692799236616637698717936, 3.31343508369343589425824123158, 4.71103900072099639482987501955, 5.57636146299207264354424341088, 6.53101822053135057434091097569, 7.18621568759072883414200492803, 8.035147353055841484261428047639, 8.846729120489478296849144119583, 9.878787543149672041354526406413

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