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Properties

Label 2-930-155.4-c1-0-13
Degree $2$
Conductor $930$
Sign $0.391 - 0.919i$
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  + (−0.951 + 0.309i)2-s + (0.951 + 0.309i)3-s + (0.809 − 0.587i)4-s + (0.618 + 2.14i)5-s − 0.999·6-s + (2.12 + 2.91i)7-s + (−0.587 + 0.809i)8-s + (0.809 + 0.587i)9-s + (−1.25 − 1.85i)10-s + (3.84 − 2.79i)11-s + (0.951 − 0.309i)12-s + (1.38 + 0.451i)13-s + (−2.91 − 2.12i)14-s + (−0.0762 + 2.23i)15-s + (0.309 − 0.951i)16-s + (4.12 − 5.68i)17-s + ⋯
L(s)  = 1  + (−0.672 + 0.218i)2-s + (0.549 + 0.178i)3-s + (0.404 − 0.293i)4-s + (0.276 + 0.961i)5-s − 0.408·6-s + (0.801 + 1.10i)7-s + (−0.207 + 0.286i)8-s + (0.269 + 0.195i)9-s + (−0.395 − 0.585i)10-s + (1.16 − 0.843i)11-s + (0.274 − 0.0892i)12-s + (0.385 + 0.125i)13-s + (−0.780 − 0.566i)14-s + (−0.0196 + 0.577i)15-s + (0.0772 − 0.237i)16-s + (1.00 − 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.391 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.391 - 0.919i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.391 - 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48325 + 0.980353i\)
\(L(\frac12)\) \(\approx\) \(1.48325 + 0.980353i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 - 0.309i)T \)
3 \( 1 + (-0.951 - 0.309i)T \)
5 \( 1 + (-0.618 - 2.14i)T \)
31 \( 1 + (5.26 + 1.80i)T \)
good7 \( 1 + (-2.12 - 2.91i)T + (-2.16 + 6.65i)T^{2} \)
11 \( 1 + (-3.84 + 2.79i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-1.38 - 0.451i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-4.12 + 5.68i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.574 - 1.76i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.863 + 1.18i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.853 + 2.62i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 - 3.41iT - 37T^{2} \)
41 \( 1 + (0.0188 + 0.0580i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (10.0 - 3.27i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (-3.59 - 1.16i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (2.47 - 3.39i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.58 + 11.0i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 1.14T + 61T^{2} \)
67 \( 1 + 7.32iT - 67T^{2} \)
71 \( 1 + (0.562 + 0.408i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.0648 + 0.0892i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-10.2 - 7.42i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (15.1 - 4.92i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (11.1 - 8.13i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (4.84 + 6.67i)T + (-29.9 + 92.2i)T^{2} \)
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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

−9.891158547323402810024377512439, −9.401383325972259442005601040583, −8.569829775576563844772633553347, −7.892825674134827757934841581861, −6.91815550977399360421011617405, −6.03906100181874339583149113017, −5.20888847442961481898389199384, −3.61150227598246588360464706880, −2.69018692277096777644663542989, −1.53456016344617259020146177194, 1.23746947637753658666161693505, 1.72162846781793822655123340271, 3.63044872996870278421987222492, 4.30815898121171423310233003419, 5.55037274257825754832115769594, 6.85361817172242205844056597816, 7.54990481838922758474690343700, 8.393724048892882037233467264247, 8.974309277521537795988579661889, 9.857742631963100040979479265978

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