↑

Properties

Label 2-930-31.25-c1-0-16
Degree $2$
Conductor $930$
Sign $-0.275 + 0.961i$
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

Related objects

Downloads

Learn more

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 + (−1.5 − 2.59i)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 + (1.5 + 2.59i)14-s + 0.999·15-s + 16-s + (−1 − 1.73i)17-s + (0.499 − 0.866i)18-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 + (−0.566 − 0.981i)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 + (0.400 + 0.694i)14-s + 0.258·15-s + 0.250·16-s + (−0.242 − 0.420i)17-s + (0.117 − 0.204i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.275 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.275 + 0.961i$
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: \((2,\ 930,\ (\ :1/2),\ -0.275 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.459294 - 0.609122i\)
\(L(\frac12)\) \(\approx\) \(0.459294 - 0.609122i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.5 + 4.33i)T \)
good7 \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 7T + 29T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 + 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + (2.5 - 4.33i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.5 - 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3 + 5.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 5T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.752915314096384660635689240277, −9.142657534957722120597509163953, −8.282725662691111679503335049537, −7.40800210902523445082908965929, −6.56099427724685621792942273860, −5.52310704239098164005590905895, −4.29530707641256811094705599474, −3.42660656730721717620560887502, −2.07000083139532324582035261716, −0.41405623119766706184646915098, 1.70649617039221832800780428782, 2.61755259340189275410882636014, 3.70186392474151333235598383124, 5.38111751583812566778080204148, 6.33257837740565716810340677601, 6.83945226507378624630101112173, 8.015751975153089056442500790047, 8.557957725351166902086562166440, 9.554737234997355804079623133824, 9.990708414421564382708284190724

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