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Properties

Label 2-930-93.26-c1-0-24
Degree $2$
Conductor $930$
Sign $0.950 + 0.310i$
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 + (0.475 − 1.66i)3-s − 4-s + (−0.866 + 0.5i)5-s + (1.66 + 0.475i)6-s + (−1.64 + 2.85i)7-s − i·8-s + (−2.54 − 1.58i)9-s + (−0.5 − 0.866i)10-s + (−0.264 − 0.458i)11-s + (−0.475 + 1.66i)12-s + (2.90 − 1.68i)13-s + (−2.85 − 1.64i)14-s + (0.421 + 1.68i)15-s + 16-s + (2.88 − 5.00i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.274 − 0.961i)3-s − 0.5·4-s + (−0.387 + 0.223i)5-s + (0.679 + 0.194i)6-s + (−0.622 + 1.07i)7-s − 0.353i·8-s + (−0.849 − 0.527i)9-s + (−0.158 − 0.273i)10-s + (−0.0798 − 0.138i)11-s + (−0.137 + 0.480i)12-s + (0.807 − 0.465i)13-s + (−0.762 − 0.440i)14-s + (0.108 + 0.433i)15-s + 0.250·16-s + (0.700 − 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.950 + 0.310i$
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: \((2,\ 930,\ (\ :1/2),\ 0.950 + 0.310i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36986 - 0.217956i\)
\(L(\frac12)\) \(\approx\) \(1.36986 - 0.217956i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.475 + 1.66i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (-1.14 - 5.44i)T \)
good7 \( 1 + (1.64 - 2.85i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.264 + 0.458i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.90 + 1.68i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.88 + 5.00i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.83 + 3.17i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 8.32T + 23T^{2} \)
29 \( 1 - 4.28T + 29T^{2} \)
37 \( 1 + (4.77 + 2.75i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.40 + 3.69i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.37 + 3.10i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 10.4iT - 47T^{2} \)
53 \( 1 + (-3.33 - 5.78i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.57 - 3.79i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 10.5iT - 61T^{2} \)
67 \( 1 + (-7.82 - 13.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-12.0 + 6.97i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.32 - 3.64i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.56 + 4.36i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.38 + 9.32i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 15.0T + 89T^{2} \)
97 \( 1 + 14.7T + 97T^{2} \)
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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

−9.711848007568373282580398643358, −8.747481312944159537751467405842, −8.509374547774441292362176946039, −7.15855923992996553992543053442, −6.94081599824695706560505564931, −5.76848852284971242090730502269, −5.16009189410357158138268476276, −3.35648433698711849148654122662, −2.73954513839028529101997143564, −0.78721678962385862059753797680, 1.17845596969517515613399011272, 3.01601638402864463931648624458, 3.77880880892406984314927881815, 4.37689798218481479362453143608, 5.50931344479238909236727916461, 6.67616393274809508697321136531, 7.926815481657218779698489992272, 8.554831340320853681337019061198, 9.590247836752646316533963085049, 10.06611041609688154889099208902

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