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Properties

Label 2-930-93.68-c1-0-39
Degree $2$
Conductor $930$
Sign $-0.995 + 0.0942i$
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 + (0.500 − 1.65i)3-s − 4-s + (0.866 + 0.5i)5-s + (−1.65 − 0.500i)6-s + (−1.28 − 2.22i)7-s + i·8-s + (−2.49 − 1.65i)9-s + (0.5 − 0.866i)10-s + (2.56 − 4.44i)11-s + (−0.500 + 1.65i)12-s + (4.59 + 2.65i)13-s + (−2.22 + 1.28i)14-s + (1.26 − 1.18i)15-s + 16-s + (−2.59 − 4.49i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.288 − 0.957i)3-s − 0.5·4-s + (0.387 + 0.223i)5-s + (−0.676 − 0.204i)6-s + (−0.486 − 0.842i)7-s + 0.353i·8-s + (−0.833 − 0.553i)9-s + (0.158 − 0.273i)10-s + (0.774 − 1.34i)11-s + (−0.144 + 0.478i)12-s + (1.27 + 0.735i)13-s + (−0.595 + 0.343i)14-s + (0.325 − 0.306i)15-s + 0.250·16-s + (−0.629 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.995 + 0.0942i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.995 + 0.0942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0719362 - 1.52362i\)
\(L(\frac12)\) \(\approx\) \(0.0719362 - 1.52362i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.500 + 1.65i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (-0.790 - 5.51i)T \)
good7 \( 1 + (1.28 + 2.22i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.56 + 4.44i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.59 - 2.65i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.59 + 4.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.264 + 0.458i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.03T + 23T^{2} \)
29 \( 1 + 8.44T + 29T^{2} \)
37 \( 1 + (-0.580 + 0.335i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (9.85 + 5.68i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.69 - 3.86i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.75iT - 47T^{2} \)
53 \( 1 + (-0.822 + 1.42i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-10.4 + 6.03i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 7.06iT - 61T^{2} \)
67 \( 1 + (-3.98 + 6.89i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.45 - 4.88i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.21 + 1.85i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.75 + 5.05i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.97 - 6.87i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 14.8T + 89T^{2} \)
97 \( 1 + 6.46T + 97T^{2} \)
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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

−9.457211698380346585580633169418, −8.979522308747031599083279681348, −8.164185387472139287037407169522, −6.79054967148667227497993923866, −6.58499214023689150881750153439, −5.34523419108091133310973267375, −3.74336639604709235890683045143, −3.22997268018113807645540817218, −1.80922107323469125649301231945, −0.71250293465955208733462787279, 1.97667264220241052066210975486, 3.48200763176424403676658253403, 4.28123848699858468082066457099, 5.39421194597903645741129340575, 6.01066235009127201750229493883, 6.95146128648598809528530994038, 8.302642411983471592019275443726, 8.764062292986862409016230336325, 9.547097565314970935464093795025, 10.12014758477660012174730431490

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