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Properties

Label 2-930-31.25-c1-0-2
Degree $2$
Conductor $930$
Sign $-0.571 - 0.820i$
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  − 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s − 8-s + (−0.499 + 0.866i)9-s + (0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)12-s + (−1 + 1.73i)13-s + (0.5 + 0.866i)14-s + 0.999·15-s + 16-s + (−1.64 − 2.85i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.288 − 0.499i)3-s + 0.5·4-s + (−0.223 + 0.387i)5-s + (0.204 + 0.353i)6-s + (−0.188 − 0.327i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 − 0.273i)10-s + (0.150 − 0.261i)11-s + (−0.144 − 0.249i)12-s + (−0.277 + 0.480i)13-s + (0.133 + 0.231i)14-s + 0.258·15-s + 0.250·16-s + (−0.399 − 0.691i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.571 - 0.820i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.571 - 0.820i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.126084 + 0.241474i\)
\(L(\frac12)\) \(\approx\) \(0.126084 + 0.241474i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-1.14 - 5.44i)T \)
good7 \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.64 + 2.85i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.64 - 2.85i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.29T + 23T^{2} \)
29 \( 1 + 6.29T + 29T^{2} \)
37 \( 1 + (0.645 + 1.11i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.64 - 4.58i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 11.2T + 47T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.5 - 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 3.29T + 61T^{2} \)
67 \( 1 + (0.645 - 1.11i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.29 - 12.6i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.29 - 3.96i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.14 - 7.18i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 10.2T + 97T^{2} \)
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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

−10.26662358131819448033600623916, −9.608077082042109913936668968972, −8.626454953602248949349512372140, −7.75687410178043076140792760574, −7.04358485733164412061460734172, −6.38353107943677925950269163073, −5.32211297424046772191797192797, −3.96038026120998768426769521792, −2.76887853579925944867846487861, −1.46868765705065528659553402138, 0.16675286034182692367101090418, 1.94110801735605350955286700965, 3.32610337519141172966699550193, 4.43960625838953508121481906515, 5.49137040459936073345599428862, 6.32620493763640991813724636748, 7.40310846588771260934708763319, 8.217761540986552364971056994411, 9.099826071169551749483190121421, 9.671664425722558721652465550562

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