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Properties

Label 2-930-15.2-c1-0-20
Degree $2$
Conductor $930$
Sign $0.939 - 0.343i$
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 + (−1.70 − 0.292i)3-s − 1.00i·4-s + (1.52 + 1.63i)5-s + (1.41 − 0.999i)6-s + (−1.33 − 1.33i)7-s + (0.707 + 0.707i)8-s + (2.82 + i)9-s + (−2.23 − 0.0743i)10-s − 1.30i·11-s + (−0.292 + 1.70i)12-s + (−1.74 + 1.74i)13-s + 1.89·14-s + (−2.12 − 3.23i)15-s − 1.00·16-s + (5.36 − 5.36i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.985 − 0.169i)3-s − 0.500i·4-s + (0.683 + 0.730i)5-s + (0.577 − 0.408i)6-s + (−0.506 − 0.506i)7-s + (0.250 + 0.250i)8-s + (0.942 + 0.333i)9-s + (−0.706 − 0.0234i)10-s − 0.394i·11-s + (−0.0845 + 0.492i)12-s + (−0.484 + 0.484i)13-s + 0.506·14-s + (−0.549 − 0.835i)15-s − 0.250·16-s + (1.30 − 1.30i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.343i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.939 - 0.343i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.939 - 0.343i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.890040 + 0.157560i\)
\(L(\frac12)\) \(\approx\) \(0.890040 + 0.157560i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 + 0.292i)T \)
5 \( 1 + (-1.52 - 1.63i)T \)
31 \( 1 - T \)
good7 \( 1 + (1.33 + 1.33i)T + 7iT^{2} \)
11 \( 1 + 1.30iT - 11T^{2} \)
13 \( 1 + (1.74 - 1.74i)T - 13iT^{2} \)
17 \( 1 + (-5.36 + 5.36i)T - 17iT^{2} \)
19 \( 1 - 4.34iT - 19T^{2} \)
23 \( 1 + (4.75 + 4.75i)T + 23iT^{2} \)
29 \( 1 - 5.50T + 29T^{2} \)
37 \( 1 + (2 + 2i)T + 37iT^{2} \)
41 \( 1 - 2.81iT - 41T^{2} \)
43 \( 1 + (-3.50 + 3.50i)T - 43iT^{2} \)
47 \( 1 + (-3.66 + 3.66i)T - 47iT^{2} \)
53 \( 1 + (-6.45 - 6.45i)T + 53iT^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 - 12.5T + 61T^{2} \)
67 \( 1 + (-5.52 - 5.52i)T + 67iT^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (1.87 - 1.87i)T - 73iT^{2} \)
79 \( 1 + 4.64iT - 79T^{2} \)
83 \( 1 + (-11.2 - 11.2i)T + 83iT^{2} \)
89 \( 1 - 7.16T + 89T^{2} \)
97 \( 1 + (1.49 + 1.49i)T + 97iT^{2} \)
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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

−10.05786885668701163091654604958, −9.682585375324860051997935467767, −8.301064543798784576776112732760, −7.21648115709602300454924715877, −6.82107663449987598223152973945, −5.90715261060522979530067446399, −5.27738710315499691093747251383, −3.92026290975454598106407344806, −2.39310730449912009716621859329, −0.817936992513367623890638459858, 0.914806761733182428441988964568, 2.19337970322318237132265111113, 3.66736497602734359571752599541, 4.86287874942469913564879863108, 5.65946385291470902289923377626, 6.41369537006419386616065367498, 7.58012963765626839352655712639, 8.534808995074136184218349626304, 9.554612574678951454839159680397, 9.970027703901517923436861171129

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