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Properties

Label 2-930-93.26-c1-0-37
Degree $2$
Conductor $930$
Sign $-0.996 - 0.0827i$
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 + (−0.346 + 1.69i)3-s − 4-s + (−0.866 + 0.5i)5-s + (1.69 + 0.346i)6-s + (1.71 − 2.97i)7-s + i·8-s + (−2.75 − 1.17i)9-s + (0.5 + 0.866i)10-s + (−0.402 − 0.697i)11-s + (0.346 − 1.69i)12-s + (−2.61 + 1.51i)13-s + (−2.97 − 1.71i)14-s + (−0.548 − 1.64i)15-s + 16-s + (−1.94 + 3.36i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.200 + 0.979i)3-s − 0.5·4-s + (−0.387 + 0.223i)5-s + (0.692 + 0.141i)6-s + (0.649 − 1.12i)7-s + 0.353i·8-s + (−0.919 − 0.392i)9-s + (0.158 + 0.273i)10-s + (−0.121 − 0.210i)11-s + (0.100 − 0.489i)12-s + (−0.725 + 0.418i)13-s + (−0.794 − 0.458i)14-s + (−0.141 − 0.424i)15-s + 0.250·16-s + (−0.471 + 0.816i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.996 - 0.0827i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.996 - 0.0827i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00339381 + 0.0818722i\)
\(L(\frac12)\) \(\approx\) \(0.00339381 + 0.0818722i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.346 - 1.69i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (1.97 + 5.20i)T \)
good7 \( 1 + (-1.71 + 2.97i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.402 + 0.697i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.61 - 1.51i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.94 - 3.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.37 - 2.39i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.142T + 23T^{2} \)
29 \( 1 + 7.20T + 29T^{2} \)
37 \( 1 + (9.28 + 5.36i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.634 + 0.366i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.711 + 0.410i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.88iT - 47T^{2} \)
53 \( 1 + (6.79 + 11.7i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (11.5 + 6.69i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 10.6iT - 61T^{2} \)
67 \( 1 + (0.342 + 0.593i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (9.83 - 5.67i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.721 + 0.416i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.81 - 1.62i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.934 + 1.61i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.24T + 89T^{2} \)
97 \( 1 - 16.1T + 97T^{2} \)
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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

−9.868133511468451809560156603751, −9.024199102230467054233462012927, −8.095204769951037454256434410773, −7.25250487811428482290266479343, −5.95765622474397215532442352729, −4.85261951205646803216429566104, −4.10652763299286984545745006735, −3.48631644484904663340492168777, −1.94017097973167713616326574381, −0.03771740857866988567798578907, 1.82472189653066854729135612106, 3.01301587916991266996944231037, 4.83450112282309299697681409305, 5.26028794553110595321424323424, 6.28816227424283149033018555698, 7.25547443496232470685936583143, 7.77609899568509452748154520461, 8.739362598263023153016464760429, 9.165996762453728553622261534150, 10.60347685340879658030736974280

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