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Properties

Label 2-930-155.37-c1-0-15
Degree $2$
Conductor $930$
Sign $-0.839 - 0.543i$
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 + (1.24 + 1.85i)5-s + (−0.866 + 0.500i)6-s + (−0.725 + 2.70i)7-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (−0.436 + 2.19i)10-s + (3.03 + 1.75i)11-s + (−0.965 − 0.258i)12-s + (3.02 − 0.809i)13-s + (−2.42 + 1.40i)14-s + (−2.11 + 0.718i)15-s − 1.00·16-s + (2.60 + 0.699i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.149 + 0.557i)3-s + 0.500i·4-s + (0.555 + 0.831i)5-s + (−0.353 + 0.204i)6-s + (−0.274 + 1.02i)7-s + (−0.250 + 0.250i)8-s + (−0.288 − 0.166i)9-s + (−0.137 + 0.693i)10-s + (0.914 + 0.528i)11-s + (−0.278 − 0.0747i)12-s + (0.838 − 0.224i)13-s + (−0.649 + 0.374i)14-s + (−0.546 + 0.185i)15-s − 0.250·16-s + (0.632 + 0.169i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.839 - 0.543i$
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: \((2,\ 930,\ (\ :1/2),\ -0.839 - 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.606554 + 2.05412i\)
\(L(\frac12)\) \(\approx\) \(0.606554 + 2.05412i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.24 - 1.85i)T \)
31 \( 1 + (5.51 - 0.735i)T \)
good7 \( 1 + (0.725 - 2.70i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3.03 - 1.75i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.02 + 0.809i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-2.60 - 0.699i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.48 + 2.01i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.26 + 4.26i)T + 23iT^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
37 \( 1 + (-0.0201 - 0.00540i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-5.33 + 9.23i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.367 + 1.36i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-6.73 - 6.73i)T + 47iT^{2} \)
53 \( 1 + (-6.26 + 1.68i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.98 - 3.45i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 8.20iT - 61T^{2} \)
67 \( 1 + (1.64 - 0.440i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (5.02 - 8.70i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-11.1 + 2.97i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.93 + 10.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.79 - 0.749i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 + (-5.44 - 5.44i)T + 97iT^{2} \)
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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

−10.43631385070168990180065398197, −9.314747324762430819906490914933, −9.064966137103413873217133545816, −7.69881964501676147764214282552, −6.81833399281242059683303121019, −5.79178177261987620698689250058, −5.62070858871126736231357251428, −4.09490337416138567423800303264, −3.30056366066584509742325278479, −2.11095693702345024029405130684, 0.944188539354035222600248378647, 1.73821052935062011389712745767, 3.50327841311197703127500485099, 4.11502731360121444099369621270, 5.57580734610108706481584198253, 5.94858029882487631835103601540, 7.10607238172156189967238506592, 7.976822793078161233156923710080, 9.169015306661664024145562095001, 9.662649568729067681557682450880

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