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Properties

Label 2-930-31.16-c1-0-14
Degree $2$
Conductor $930$
Sign $-0.390 + 0.920i$
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  + (−0.809 + 0.587i)2-s + (−0.809 − 0.587i)3-s + (0.309 − 0.951i)4-s − 5-s + 6-s + (−1.26 + 3.90i)7-s + (0.309 + 0.951i)8-s + (0.309 + 0.951i)9-s + (0.809 − 0.587i)10-s + (0.769 − 2.36i)11-s + (−0.809 + 0.587i)12-s + (−0.662 − 0.481i)13-s + (−1.26 − 3.90i)14-s + (0.809 + 0.587i)15-s + (−0.809 − 0.587i)16-s + (0.162 + 0.501i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.467 − 0.339i)3-s + (0.154 − 0.475i)4-s − 0.447·5-s + 0.408·6-s + (−0.479 + 1.47i)7-s + (0.109 + 0.336i)8-s + (0.103 + 0.317i)9-s + (0.255 − 0.185i)10-s + (0.232 − 0.714i)11-s + (−0.233 + 0.169i)12-s + (−0.183 − 0.133i)13-s + (−0.339 − 1.04i)14-s + (0.208 + 0.151i)15-s + (−0.202 − 0.146i)16-s + (0.0394 + 0.121i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.390 + 0.920i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.390 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.129546 - 0.195678i\)
\(L(\frac12)\) \(\approx\) \(0.129546 - 0.195678i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + T \)
31 \( 1 + (1.17 + 5.44i)T \)
good7 \( 1 + (1.26 - 3.90i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (-0.769 + 2.36i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.662 + 0.481i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.162 - 0.501i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.35 - 1.70i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.293 + 0.904i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-0.589 + 0.427i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 + 3.68T + 37T^{2} \)
41 \( 1 + (-9.06 + 6.58i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (3.80 - 2.76i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-0.555 - 0.403i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.65 + 8.16i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (10.7 + 7.78i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + 14.4T + 61T^{2} \)
67 \( 1 + 8.88T + 67T^{2} \)
71 \( 1 + (3.98 + 12.2i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.0958 + 0.294i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-0.715 - 2.20i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.81 - 2.77i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-2.01 + 6.20i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-5.06 + 15.5i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.608899648816382574757360331684, −8.874552511804862480567876878328, −8.191355664000152904501333911286, −7.32450018197698045064083597835, −6.09591731251580222677971021803, −5.96315843118669339676971309102, −4.71742952039287254684527783632, −3.21755500681510148771612444127, −1.96690041089795765570622241324, −0.15185818059642621798995167584, 1.28534684790867508801984459559, 3.04624775756078080635258034876, 4.09948226481740522170051130755, 4.69473688717223647141009320541, 6.28099708165377480304290019762, 7.14127796336691352071087896587, 7.63772186703500988309477389315, 8.904949204944866734406088534061, 9.629919848831809897137361113861, 10.51329110271199373342875051565

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