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Properties

Label 2-930-155.37-c1-0-28
Degree $2$
Conductor $930$
Sign $-0.955 + 0.295i$
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  + (−0.707 − 0.707i)2-s + (0.258 − 0.965i)3-s + 1.00i·4-s + (2.11 − 0.714i)5-s + (−0.866 + 0.500i)6-s + (0.511 − 1.90i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (−2.00 − 0.993i)10-s + (−1.57 − 0.908i)11-s + (0.965 + 0.258i)12-s + (−3.83 + 1.02i)13-s + (−1.71 + 0.987i)14-s + (−0.141 − 2.23i)15-s − 1.00·16-s + (−7.08 − 1.89i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.149 − 0.557i)3-s + 0.500i·4-s + (0.947 − 0.319i)5-s + (−0.353 + 0.204i)6-s + (0.193 − 0.720i)7-s + (0.250 − 0.250i)8-s + (−0.288 − 0.166i)9-s + (−0.633 − 0.314i)10-s + (−0.474 − 0.274i)11-s + (0.278 + 0.0747i)12-s + (−1.06 + 0.285i)13-s + (−0.457 + 0.263i)14-s + (−0.0365 − 0.576i)15-s − 0.250·16-s + (−1.71 − 0.460i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.955 + 0.295i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.955 + 0.295i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.158932 - 1.05153i\)
\(L(\frac12)\) \(\approx\) \(0.158932 - 1.05153i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (-2.11 + 0.714i)T \)
31 \( 1 + (5.31 + 1.64i)T \)
good7 \( 1 + (-0.511 + 1.90i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.57 + 0.908i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.83 - 1.02i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (7.08 + 1.89i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.55 + 1.47i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.62 + 2.62i)T + 23iT^{2} \)
29 \( 1 - 6.71T + 29T^{2} \)
37 \( 1 + (-3.80 - 1.01i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.97 + 3.41i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.271 - 1.01i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (7.79 + 7.79i)T + 47iT^{2} \)
53 \( 1 + (-3.83 + 1.02i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.05 + 2.91i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 1.54iT - 61T^{2} \)
67 \( 1 + (6.17 - 1.65i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-5.23 + 9.07i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (12.5 - 3.36i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-8.71 - 15.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-16.5 + 4.42i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 17.9T + 89T^{2} \)
97 \( 1 + (-9.33 - 9.33i)T + 97iT^{2} \)
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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

−9.676908352872490323664188816822, −8.957128931434310154550391546367, −8.139708636668200098552938054831, −7.15799488760640880548530611444, −6.53617177754877121047007686916, −5.21004248734998075783161558964, −4.32145813433938740699160151137, −2.71820224256295046032653250645, −2.00035577385476347700947744034, −0.53977302823208354730702485883, 1.96534480457222805273395454334, 2.81543901796875565573365144858, 4.52725044344820771015684933279, 5.34950001528187061750629011459, 6.10217605809920998838962562291, 7.07094548799660718660809922633, 8.034955381247562863713421588603, 8.933330875404215317267032827443, 9.549986083022765158434401112677, 10.22678997475563495599259392062

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