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Properties

Label 2-930-155.37-c1-0-27
Degree $2$
Conductor $930$
Sign $0.552 + 0.833i$
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 + (1.20 − 1.88i)5-s + (0.866 − 0.500i)6-s + (0.614 − 2.29i)7-s + (−0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + (2.18 − 0.482i)10-s + (−1.25 − 0.724i)11-s + (0.965 + 0.258i)12-s + (0.0155 − 0.00418i)13-s + (2.05 − 1.18i)14-s + (−1.50 − 1.64i)15-s − 1.00·16-s + (4.54 + 1.21i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.149 − 0.557i)3-s + 0.500i·4-s + (0.537 − 0.843i)5-s + (0.353 − 0.204i)6-s + (0.232 − 0.867i)7-s + (−0.250 + 0.250i)8-s + (−0.288 − 0.166i)9-s + (0.690 − 0.152i)10-s + (−0.378 − 0.218i)11-s + (0.278 + 0.0747i)12-s + (0.00432 − 0.00115i)13-s + (0.549 − 0.317i)14-s + (−0.389 − 0.425i)15-s − 0.250·16-s + (1.10 + 0.295i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.552 + 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.552 + 0.833i$
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: \((2,\ 930,\ (\ :1/2),\ 0.552 + 0.833i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99397 - 1.07113i\)
\(L(\frac12)\) \(\approx\) \(1.99397 - 1.07113i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.20 + 1.88i)T \)
31 \( 1 + (-3.55 + 4.28i)T \)
good7 \( 1 + (-0.614 + 2.29i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (1.25 + 0.724i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.0155 + 0.00418i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-4.54 - 1.21i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.111 - 0.0644i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.84 + 4.84i)T + 23iT^{2} \)
29 \( 1 - 4.76T + 29T^{2} \)
37 \( 1 + (8.93 + 2.39i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.12 - 3.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.846 + 3.16i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-3.51 - 3.51i)T + 47iT^{2} \)
53 \( 1 + (-11.0 + 2.96i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (10.4 - 6.02i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 7.91iT - 61T^{2} \)
67 \( 1 + (-5.92 + 1.58i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.242 - 0.420i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.88 + 0.772i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.94 - 6.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-13.8 + 3.72i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 2.63T + 89T^{2} \)
97 \( 1 + (-6.26 - 6.26i)T + 97iT^{2} \)
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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

−10.00095515196441248312670210673, −8.790362246063400054642684610146, −8.132694556326229984412527660349, −7.45542486042723860525255810201, −6.39694844378146828130578069232, −5.66625715825085397312911893148, −4.71167194966070479266656070619, −3.76888933464428793644005529231, −2.35419620647059878117179501436, −0.922726219519993809041148394870, 1.87452886140887146699514486958, 2.86186438463134279236105584957, 3.65986877871022206665596888154, 5.06396781510850285360284244474, 5.58305744756273308750847064312, 6.56172122492869288349869342343, 7.69647311304994157367518137183, 8.723019467091842342924513640770, 9.689745367520598923591203723011, 10.20822191977690784853888897842

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