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Properties

Label 2-930-93.26-c1-0-43
Degree $2$
Conductor $930$
Sign $-0.222 - 0.975i$
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 + (0.645 − 1.60i)3-s − 4-s + (0.866 − 0.5i)5-s + (−1.60 − 0.645i)6-s + (−2.35 + 4.08i)7-s + i·8-s + (−2.16 − 2.07i)9-s + (−0.5 − 0.866i)10-s + (−3.18 − 5.51i)11-s + (−0.645 + 1.60i)12-s + (−2.96 + 1.71i)13-s + (4.08 + 2.35i)14-s + (−0.244 − 1.71i)15-s + 16-s + (−1.65 + 2.87i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.372 − 0.928i)3-s − 0.5·4-s + (0.387 − 0.223i)5-s + (−0.656 − 0.263i)6-s + (−0.890 + 1.54i)7-s + 0.353i·8-s + (−0.722 − 0.691i)9-s + (−0.158 − 0.273i)10-s + (−0.960 − 1.66i)11-s + (−0.186 + 0.464i)12-s + (−0.823 + 0.475i)13-s + (1.09 + 0.629i)14-s + (−0.0632 − 0.442i)15-s + 0.250·16-s + (−0.402 + 0.696i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.222 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.222 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.222 - 0.975i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.222 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0672838 + 0.0843331i\)
\(L(\frac12)\) \(\approx\) \(0.0672838 + 0.0843331i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.645 + 1.60i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (-3.67 + 4.18i)T \)
good7 \( 1 + (2.35 - 4.08i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.18 + 5.51i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.96 - 1.71i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.65 - 2.87i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.56 - 4.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.89T + 23T^{2} \)
29 \( 1 + 5.57T + 29T^{2} \)
37 \( 1 + (4.26 + 2.46i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (8.33 - 4.81i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.26 + 3.04i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.50iT - 47T^{2} \)
53 \( 1 + (0.645 + 1.11i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.87 - 2.81i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 1.31iT - 61T^{2} \)
67 \( 1 + (-7.24 - 12.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.61 + 0.929i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-9.05 + 5.22i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.954 - 0.551i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.19 + 5.54i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.30T + 89T^{2} \)
97 \( 1 - 0.469T + 97T^{2} \)
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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

−9.361688313456697793767184229994, −8.578934663852447789978065286094, −8.260863469818985983270459003080, −6.76872419500676067383085513695, −5.87601636584778677934484112540, −5.36188240425726337677262054797, −3.52054410769315308921300217652, −2.69523306805172139117949039301, −1.91713561669057646016777634617, −0.04387656869901756866960531454, 2.51662744453653316119130503386, 3.57149120873460665224511552075, 4.83272933206604909315698521438, 5.00438238468169601223925932878, 6.79714046628810875311486430465, 7.08847555323134951852891823636, 8.009095289811465092994346946776, 9.254461113079039227576854223053, 9.822563666513387420208602542377, 10.32280235316707280035178739144

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