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Properties

Label 2-930-155.39-c1-0-7
Degree $2$
Conductor $930$
Sign $-0.820 - 0.571i$
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 + (−0.0762 + 2.23i)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 + (−1.25 − 1.85i)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 + (−0.0241 + 0.706i)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 + (−0.323 − 0.478i)15-s + (0.0772 + 0.237i)16-s + (−1.00 − 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 - 0.571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.820 - 0.571i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.820 - 0.571i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.483343 + 1.54070i\)
\(L(\frac12)\) \(\approx\) \(0.483343 + 1.54070i\)
\(L(\frac{3}{2})\) 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

−10.51513350571622302192660626330, −9.365405694109204574487837806708, −9.191918338104660912936199073598, −7.37414580317798004100272437206, −6.76877522324121149219734185203, −6.20298509972093189050536973722, −5.24469895121470648280173149105, −4.27568041515923525220506041930, −3.06575761240349915509997424776, −2.18456845924310899808395437961, 0.63568357458544426486500494220, 1.88242784866822246254436508480, 3.79927165778891674158978818632, 4.09800896182925407270939763730, 5.43344315284991521924014002466, 6.17835050322977073021192755410, 6.83898102371249422374370130462, 7.968444325997239993194289965422, 9.065169782081247123249507437087, 9.837226020662445351430525267032

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