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Properties

Label 2-930-155.37-c1-0-5
Degree $2$
Conductor $930$
Sign $-0.718 - 0.695i$
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 + (0.303 − 2.21i)5-s + (0.866 − 0.500i)6-s + (−1.35 + 5.06i)7-s + (0.707 − 0.707i)8-s + (−0.866 − 0.499i)9-s + (−1.78 + 1.35i)10-s + (4.89 + 2.82i)11-s + (−0.965 − 0.258i)12-s + (−2.98 + 0.799i)13-s + (4.54 − 2.62i)14-s + (2.06 + 0.866i)15-s − 1.00·16-s + (−6.10 − 1.63i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.149 + 0.557i)3-s + 0.500i·4-s + (0.135 − 0.990i)5-s + (0.353 − 0.204i)6-s + (−0.512 + 1.91i)7-s + (0.250 − 0.250i)8-s + (−0.288 − 0.166i)9-s + (−0.563 + 0.427i)10-s + (1.47 + 0.852i)11-s + (−0.278 − 0.0747i)12-s + (−0.827 + 0.221i)13-s + (1.21 − 0.700i)14-s + (0.532 + 0.223i)15-s − 0.250·16-s + (−1.48 − 0.397i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.718 - 0.695i$
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: \((2,\ 930,\ (\ :1/2),\ -0.718 - 0.695i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.188128 + 0.464463i\)
\(L(\frac12)\) \(\approx\) \(0.188128 + 0.464463i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.303 + 2.21i)T \)
31 \( 1 + (4.39 - 3.42i)T \)
good7 \( 1 + (1.35 - 5.06i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-4.89 - 2.82i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.98 - 0.799i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (6.10 + 1.63i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.97 + 1.14i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.69 + 4.69i)T + 23iT^{2} \)
29 \( 1 - 2.75T + 29T^{2} \)
37 \( 1 + (-3.09 - 0.828i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (4.15 - 7.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.62 - 6.08i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.83 + 1.83i)T + 47iT^{2} \)
53 \( 1 + (-6.83 + 1.83i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (6.91 - 3.99i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 2.01iT - 61T^{2} \)
67 \( 1 + (-4.78 + 1.28i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (6.93 - 12.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (16.3 - 4.37i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-1.90 - 3.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.65 - 2.05i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 1.97T + 89T^{2} \)
97 \( 1 + (2.48 + 2.48i)T + 97iT^{2} \)
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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

−9.920864533568700649465823468824, −9.548945921695955167351050124039, −8.886564715280497220744216211627, −8.473059907249844483778166189098, −6.89085714218724303408415194250, −6.06576394685698176159641208835, −4.90678612987779249874947523665, −4.28721633113663685990341881766, −2.75588258755750173187976863051, −1.82469504644244533694855971804, 0.27348834340127636960960714457, 1.74201488049439151480654573721, 3.42212635662385984217869889093, 4.22135934569847732795996702454, 5.91586969407721385843065379910, 6.54822742868101767111492816649, 7.19930540977447678736446489980, 7.67836416809606910207791275901, 8.930213684440583037100869015178, 9.830706070716301944146626092433

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