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Properties

Label 2-930-15.8-c1-0-21
Degree $2$
Conductor $930$
Sign $0.993 + 0.115i$
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 + (−1.73 − 0.0809i)3-s + 1.00i·4-s + (−0.662 + 2.13i)5-s + (1.16 + 1.28i)6-s + (2.89 − 2.89i)7-s + (0.707 − 0.707i)8-s + (2.98 + 0.280i)9-s + (1.97 − 1.04i)10-s + 1.62i·11-s + (0.0809 − 1.73i)12-s + (−1.74 − 1.74i)13-s − 4.09·14-s + (1.31 − 3.64i)15-s − 1.00·16-s + (−0.396 − 0.396i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.998 − 0.0467i)3-s + 0.500i·4-s + (−0.296 + 0.955i)5-s + (0.476 + 0.522i)6-s + (1.09 − 1.09i)7-s + (0.250 − 0.250i)8-s + (0.995 + 0.0933i)9-s + (0.625 − 0.329i)10-s + 0.489i·11-s + (0.0233 − 0.499i)12-s + (−0.484 − 0.484i)13-s − 1.09·14-s + (0.340 − 0.940i)15-s − 0.250·16-s + (−0.0960 − 0.0960i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.993 + 0.115i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.993 + 0.115i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.879065 - 0.0508217i\)
\(L(\frac12)\) \(\approx\) \(0.879065 - 0.0508217i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 0.0809i)T \)
5 \( 1 + (0.662 - 2.13i)T \)
31 \( 1 + T \)
good7 \( 1 + (-2.89 + 2.89i)T - 7iT^{2} \)
11 \( 1 - 1.62iT - 11T^{2} \)
13 \( 1 + (1.74 + 1.74i)T + 13iT^{2} \)
17 \( 1 + (0.396 + 0.396i)T + 17iT^{2} \)
19 \( 1 - 4.55iT - 19T^{2} \)
23 \( 1 + (-0.0718 + 0.0718i)T - 23iT^{2} \)
29 \( 1 - 3.20T + 29T^{2} \)
37 \( 1 + (-5.51 + 5.51i)T - 37iT^{2} \)
41 \( 1 + 4.20iT - 41T^{2} \)
43 \( 1 + (-7.67 - 7.67i)T + 43iT^{2} \)
47 \( 1 + (-6.63 - 6.63i)T + 47iT^{2} \)
53 \( 1 + (2.49 - 2.49i)T - 53iT^{2} \)
59 \( 1 - 6.90T + 59T^{2} \)
61 \( 1 - 8.74T + 61T^{2} \)
67 \( 1 + (-7.82 + 7.82i)T - 67iT^{2} \)
71 \( 1 - 7.38iT - 71T^{2} \)
73 \( 1 + (-7.37 - 7.37i)T + 73iT^{2} \)
79 \( 1 - 0.229iT - 79T^{2} \)
83 \( 1 + (0.833 - 0.833i)T - 83iT^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 + (-7.96 + 7.96i)T - 97iT^{2} \)
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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

−10.26062481727591434916347649359, −9.673680265624239360898300885952, −8.063132352053299142641201494466, −7.53700264274603129198728660252, −6.91502830039217326364882337971, −5.74999483260430734916561730301, −4.53636978334063707353546404382, −3.86138872517798852883751877253, −2.28573676272570803257088164392, −0.932676123804960452286185629234, 0.801968697044977789757331927630, 2.13361055535566443553945144265, 4.30187301473205308862749018364, 5.07068763357172430637115450630, 5.58812717596845318134705312495, 6.63087037387948042885568684550, 7.62759063233076396626430154286, 8.504086518741235727176844772392, 9.054936076227327597305636473342, 9.954526128138349722649678071670

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