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Properties

Label 2-930-93.68-c1-0-0
Degree $2$
Conductor $930$
Sign $-0.0667 - 0.997i$
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 + (−1.57 + 0.714i)3-s − 4-s + (−0.866 − 0.5i)5-s + (0.714 + 1.57i)6-s + (−2.21 − 3.83i)7-s + i·8-s + (1.97 − 2.25i)9-s + (−0.5 + 0.866i)10-s + (−0.0757 + 0.131i)11-s + (1.57 − 0.714i)12-s + (−0.295 − 0.170i)13-s + (−3.83 + 2.21i)14-s + (1.72 + 0.170i)15-s + 16-s + (−3.50 − 6.07i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.911 + 0.412i)3-s − 0.5·4-s + (−0.387 − 0.223i)5-s + (0.291 + 0.644i)6-s + (−0.837 − 1.45i)7-s + 0.353i·8-s + (0.659 − 0.751i)9-s + (−0.158 + 0.273i)10-s + (−0.0228 + 0.0395i)11-s + (0.455 − 0.206i)12-s + (−0.0818 − 0.0472i)13-s + (−1.02 + 0.592i)14-s + (0.445 + 0.0440i)15-s + 0.250·16-s + (−0.851 − 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0667 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0667 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.0667 - 0.997i$
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: \((2,\ 930,\ (\ :1/2),\ -0.0667 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0297247 + 0.0317789i\)
\(L(\frac12)\) \(\approx\) \(0.0297247 + 0.0317789i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.57 - 0.714i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 + (3.15 - 4.58i)T \)
good7 \( 1 + (2.21 + 3.83i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.0757 - 0.131i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.295 + 0.170i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.50 + 6.07i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.09 - 5.36i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.85T + 23T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
37 \( 1 + (3.45 - 1.99i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.0353 - 0.0203i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.78 + 3.91i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.796iT - 47T^{2} \)
53 \( 1 + (6.46 - 11.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.54 - 0.891i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 12.6iT - 61T^{2} \)
67 \( 1 + (3.90 - 6.75i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.62 + 3.82i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.70 - 2.71i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-13.6 + 7.90i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.37 + 2.38i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 0.746T + 89T^{2} \)
97 \( 1 + 6.38T + 97T^{2} \)
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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

−10.54019600708967243296601820387, −9.558720224074811860889050187653, −9.122418696356829090717227299636, −7.42915247153202169133885511735, −7.08820090195938901667631397892, −5.78079854886199092799855778172, −4.80467727571741671378296960630, −3.97276483230658944184407493127, −3.21945475572028115811369298890, −1.16014323184581504324746063824, 0.02635550952895792306901477055, 2.14595443069869126975735818588, 3.57597769894841791898118980784, 4.90363594827968593919164789104, 5.69260342234005611209842234779, 6.38643722580459467906675853355, 7.07880687085288964293014654235, 8.014400459163497697766796018532, 9.053232603025808616184179419011, 9.572605302514874776313283510367

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