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Properties

Label 2-930-155.27-c1-0-7
Degree $2$
Conductor $930$
Sign $0.611 - 0.790i$
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  + (0.453 − 0.891i)2-s + (−0.891 + 0.453i)3-s + (−0.587 − 0.809i)4-s + (−0.0743 + 2.23i)5-s + 1.00i·6-s + (2.80 + 0.444i)7-s + (−0.987 + 0.156i)8-s + (0.587 − 0.809i)9-s + (1.95 + 1.08i)10-s + (0.168 + 0.232i)11-s + (0.891 + 0.453i)12-s + (−2.26 + 1.15i)13-s + (1.66 − 2.29i)14-s + (−0.948 − 2.02i)15-s + (−0.309 + 0.951i)16-s + (3.46 − 0.549i)17-s + ⋯
L(s)  = 1  + (0.321 − 0.630i)2-s + (−0.514 + 0.262i)3-s + (−0.293 − 0.404i)4-s + (−0.0332 + 0.999i)5-s + 0.408i·6-s + (1.06 + 0.167i)7-s + (−0.349 + 0.0553i)8-s + (0.195 − 0.269i)9-s + (0.619 + 0.341i)10-s + (0.0508 + 0.0700i)11-s + (0.257 + 0.131i)12-s + (−0.628 + 0.320i)13-s + (0.446 − 0.614i)14-s + (−0.244 − 0.522i)15-s + (−0.0772 + 0.237i)16-s + (0.841 − 0.133i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 - 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.611 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.611 - 0.790i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.611 - 0.790i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29177 + 0.633792i\)
\(L(\frac12)\) \(\approx\) \(1.29177 + 0.633792i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.453 + 0.891i)T \)
3 \( 1 + (0.891 - 0.453i)T \)
5 \( 1 + (0.0743 - 2.23i)T \)
31 \( 1 + (-4.58 - 3.15i)T \)
good7 \( 1 + (-2.80 - 0.444i)T + (6.65 + 2.16i)T^{2} \)
11 \( 1 + (-0.168 - 0.232i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (2.26 - 1.15i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (-3.46 + 0.549i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (5.50 - 1.78i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.758 - 4.78i)T + (-21.8 + 7.10i)T^{2} \)
29 \( 1 + (-1.12 - 3.46i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + (2.05 - 2.05i)T - 37iT^{2} \)
41 \( 1 + (-2.08 - 6.41i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-0.0455 + 0.0893i)T + (-25.2 - 34.7i)T^{2} \)
47 \( 1 + (-4.83 + 2.46i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (0.0146 + 0.0926i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (-12.6 - 4.10i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 - 8.78iT - 61T^{2} \)
67 \( 1 + (1.63 + 1.63i)T + 67iT^{2} \)
71 \( 1 + (8.47 + 6.15i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.79 + 0.601i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-11.1 - 8.07i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.645 + 1.26i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (-7.19 + 5.22i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (15.0 + 2.38i)T + (92.2 + 29.9i)T^{2} \)
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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

−10.38612550633285195018942398547, −9.732091703871934980916204991977, −8.588453128707412486238156899703, −7.60226036134338541605396585073, −6.67246920317749109514767721077, −5.67930035257228573665153115993, −4.83715613480156626590833454522, −3.90941498670816745276519438799, −2.75922419240981527406170959917, −1.56989203610187536135128539050, 0.68239906866750262061583471713, 2.23099122623081771270777191591, 4.11148156071390936118774147159, 4.77583835671209333498527260896, 5.49651228457558455014126616939, 6.40360551710824179945004651832, 7.50030333210374609136392208235, 8.150482097975785914262510740627, 8.790025047645744880094131379097, 9.938450598721222037306726590273

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