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Properties

Label 2-930-93.68-c1-0-38
Degree $2$
Conductor $930$
Sign $-0.543 + 0.839i$
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  − i·2-s + (1.62 − 0.595i)3-s − 4-s + (0.866 + 0.5i)5-s + (−0.595 − 1.62i)6-s + (−0.581 − 1.00i)7-s + i·8-s + (2.29 − 1.93i)9-s + (0.5 − 0.866i)10-s + (0.0889 − 0.154i)11-s + (−1.62 + 0.595i)12-s + (−4.88 − 2.82i)13-s + (−1.00 + 0.581i)14-s + (1.70 + 0.297i)15-s + 16-s + (−1.93 − 3.35i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.939 − 0.343i)3-s − 0.5·4-s + (0.387 + 0.223i)5-s + (−0.243 − 0.664i)6-s + (−0.219 − 0.380i)7-s + 0.353i·8-s + (0.763 − 0.645i)9-s + (0.158 − 0.273i)10-s + (0.0268 − 0.0464i)11-s + (−0.469 + 0.171i)12-s + (−1.35 − 0.782i)13-s + (−0.269 + 0.155i)14-s + (0.440 + 0.0768i)15-s + 0.250·16-s + (−0.470 − 0.814i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.543 + 0.839i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.543 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.932894 - 1.71518i\)
\(L(\frac12)\) \(\approx\) \(0.932894 - 1.71518i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-1.62 + 0.595i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (-5.12 + 2.17i)T \)
good7 \( 1 + (0.581 + 1.00i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.0889 + 0.154i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.88 + 2.82i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.93 + 3.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.57 + 4.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.70T + 23T^{2} \)
29 \( 1 - 5.56T + 29T^{2} \)
37 \( 1 + (-0.782 + 0.451i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.48 - 1.43i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.735 + 0.424i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.11iT - 47T^{2} \)
53 \( 1 + (6.25 - 10.8i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.27 - 2.46i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 0.00227iT - 61T^{2} \)
67 \( 1 + (3.20 - 5.54i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-10.9 - 6.33i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-9.32 - 5.38i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.52 - 1.45i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.95 - 13.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.20T + 89T^{2} \)
97 \( 1 + 13.7T + 97T^{2} \)
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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

−9.750591596065717226488982300668, −9.140694819049760170766061472866, −8.243419755888133615139986296189, −7.25723287589292601486290809938, −6.64127300638254234243421034009, −5.11922257569887909942926983147, −4.25607496521736624927778216242, −2.80594325541172491057226596815, −2.57322937006457569765076758777, −0.835206373593826638735667987710, 1.86511012297786180771984036813, 3.00675370869894690669530105706, 4.33854820522785715750703932875, 4.93207272745034112090720882886, 6.20004519638928716328126980669, 6.97178580010708914375739215217, 7.987775641138563928746970232762, 8.650827744839402233606934719205, 9.417191141029594767471452970428, 9.959273663066960582272203980360

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