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Properties

Label 2-930-31.16-c1-0-3
Degree $2$
Conductor $930$
Sign $-0.0388 - 0.999i$
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 + (0.0931 − 0.286i)7-s + (0.309 + 0.951i)8-s + (0.309 + 0.951i)9-s + (0.809 − 0.587i)10-s + (−0.593 + 1.82i)11-s + (−0.809 + 0.587i)12-s + (1.37 + 0.997i)13-s + (0.0931 + 0.286i)14-s + (0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (−1.87 − 5.76i)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 + (0.0352 − 0.108i)7-s + (0.109 + 0.336i)8-s + (0.103 + 0.317i)9-s + (0.255 − 0.185i)10-s + (−0.178 + 0.550i)11-s + (−0.233 + 0.169i)12-s + (0.380 + 0.276i)13-s + (0.0248 + 0.0766i)14-s + (0.208 + 0.151i)15-s + (−0.202 − 0.146i)16-s + (−0.454 − 1.39i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0388 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0388 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.0388 - 0.999i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.0388 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.437292 + 0.454634i\)
\(L(\frac12)\) \(\approx\) \(0.437292 + 0.454634i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.38 - 4.42i)T \)
good7 \( 1 + (-0.0931 + 0.286i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (0.593 - 1.82i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.37 - 0.997i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.87 + 5.76i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.820 - 0.596i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.226 - 0.697i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (6.77 - 4.92i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 - 5.63T + 37T^{2} \)
41 \( 1 + (8.62 - 6.26i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (-3.50 + 2.54i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-6.97 - 5.06i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.79 - 11.6i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-10.0 - 7.26i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + 4.04T + 61T^{2} \)
67 \( 1 + 8.33T + 67T^{2} \)
71 \( 1 + (4.12 + 12.6i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.20 - 12.9i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.36 - 13.4i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.46 - 3.24i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (1.55 - 4.77i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-3.23 + 9.96i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.29980133826237412294972477430, −9.310174107279513366210872749128, −8.632892378406374664047682897839, −7.45584068027279720031261144209, −7.17890390421823974869753574400, −6.14439386053449430800992976829, −5.15217339285353284725208776705, −4.24191388980049273661535648316, −2.66397034949611589734120704719, −1.18490246431982434489720430461, 0.43438556903692472001332084362, 2.11041143285199463116146997403, 3.54073473293981800525333172408, 4.24675432734222070435809698859, 5.59359686336464673839614665403, 6.37326237045126323862811283761, 7.50529751964003268128369883196, 8.376730130793715927904784546710, 8.949711657874704362443307843115, 10.08614124472866369414288556973

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