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Properties

Label 2-930-93.68-c1-0-24
Degree $2$
Conductor $930$
Sign $0.688 - 0.725i$
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 + (−0.449 + 1.67i)3-s − 4-s + (−0.866 − 0.5i)5-s + (−1.67 − 0.449i)6-s + (−1.32 − 2.29i)7-s − i·8-s + (−2.59 − 1.50i)9-s + (0.5 − 0.866i)10-s + (1.35 − 2.34i)11-s + (0.449 − 1.67i)12-s + (3.49 + 2.01i)13-s + (2.29 − 1.32i)14-s + (1.22 − 1.22i)15-s + 16-s + (−0.370 − 0.642i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.259 + 0.965i)3-s − 0.5·4-s + (−0.387 − 0.223i)5-s + (−0.682 − 0.183i)6-s + (−0.499 − 0.865i)7-s − 0.353i·8-s + (−0.865 − 0.500i)9-s + (0.158 − 0.273i)10-s + (0.408 − 0.707i)11-s + (0.129 − 0.482i)12-s + (0.970 + 0.560i)13-s + (0.612 − 0.353i)14-s + (0.316 − 0.316i)15-s + 0.250·16-s + (−0.0899 − 0.155i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.688 - 0.725i$
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: \((2,\ 930,\ (\ :1/2),\ 0.688 - 0.725i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06280 + 0.456687i\)
\(L(\frac12)\) \(\approx\) \(1.06280 + 0.456687i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.449 - 1.67i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 + (-4.10 - 3.76i)T \)
good7 \( 1 + (1.32 + 2.29i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.35 + 2.34i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.49 - 2.01i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.370 + 0.642i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.17 - 2.03i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.17T + 23T^{2} \)
29 \( 1 + 1.11T + 29T^{2} \)
37 \( 1 + (-8.66 + 5.00i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.56 + 2.63i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.20 + 2.42i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.29iT - 47T^{2} \)
53 \( 1 + (0.432 - 0.748i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.0673 + 0.0388i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 8.21iT - 61T^{2} \)
67 \( 1 + (-4.09 + 7.08i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (11.6 + 6.71i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-9.24 - 5.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.5 - 6.09i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.44 + 4.24i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 17.1T + 89T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−10.09797645151645357480131773146, −9.190590974403720246851683318237, −8.668752394499074771453441069622, −7.64819388828514148249059991199, −6.59035915882072667383248468949, −5.94956506098427408810421103760, −4.84198413127569430300423948058, −3.95051095180909483722558274759, −3.36087243210647291602321742509, −0.76250513066620287696075717438, 1.02926097771991260307374605362, 2.40620449522374256177627748859, 3.23592071278256361107373626953, 4.57654171642000510766649289140, 5.75908330615852088721149156756, 6.45831147842210180532452134541, 7.46509932630017085305933780964, 8.366872359504777904129773529031, 9.082789569338655772856692236096, 10.03838297384129787730686283245

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