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Properties

Label 2-930-31.5-c1-0-3
Degree $2$
Conductor $930$
Sign $0.968 - 0.249i$
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  − 2-s + (0.5 − 0.866i)3-s + 4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (0.954 − 1.65i)7-s − 8-s + (−0.499 − 0.866i)9-s + (0.5 + 0.866i)10-s + (3.18 + 5.52i)11-s + (0.5 − 0.866i)12-s + (2.90 + 5.03i)13-s + (−0.954 + 1.65i)14-s − 0.999·15-s + 16-s + (−3.45 + 5.98i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.288 − 0.499i)3-s + 0.5·4-s + (−0.223 − 0.387i)5-s + (−0.204 + 0.353i)6-s + (0.360 − 0.625i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 + 0.273i)10-s + (0.961 + 1.66i)11-s + (0.144 − 0.249i)12-s + (0.806 + 1.39i)13-s + (−0.255 + 0.441i)14-s − 0.258·15-s + 0.250·16-s + (−0.837 + 1.45i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.968 - 0.249i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.968 - 0.249i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26847 + 0.161024i\)
\(L(\frac12)\) \(\approx\) \(1.26847 + 0.161024i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-4.42 + 3.38i)T \)
good7 \( 1 + (-0.954 + 1.65i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.18 - 5.52i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.90 - 5.03i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.45 - 5.98i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.22 - 2.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.909T + 23T^{2} \)
29 \( 1 - 5.46T + 29T^{2} \)
37 \( 1 + (-4.45 + 7.71i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.59 - 6.23i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.232 + 0.402i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.72T + 47T^{2} \)
53 \( 1 + (3.64 + 6.30i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.86 + 4.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 4.44T + 61T^{2} \)
67 \( 1 + (4.45 + 7.71i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.90 - 8.50i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.232 + 0.402i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.32 - 10.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 - 18.1T + 97T^{2} \)
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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

−9.878778689182870435526776673210, −9.226954804737412967209675744400, −8.382426143133843028346424563306, −7.74761238287074654467081209618, −6.67840224586799109271552347040, −6.35300283046603764440991804377, −4.42077045622630873325299285961, −4.00727056858334276845732019987, −2.03415528666099291936647266846, −1.38199336531307798293077722610, 0.830819713454785397026578658730, 2.72037108129033903323196737598, 3.32104770820329148273444283673, 4.72662315502910275863687209632, 5.89213959762560394313143012511, 6.58378041927862213350024310648, 7.82377474220366977751215518354, 8.652940293341889137627966855117, 8.883389124192103513647190843667, 10.02631546155349656981316548876

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