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Properties

Label 2-930-93.68-c1-0-22
Degree $2$
Conductor $930$
Sign $0.747 - 0.664i$
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 + (1.29 + 1.14i)3-s − 4-s + (0.866 + 0.5i)5-s + (1.14 − 1.29i)6-s + (1.71 + 2.97i)7-s + i·8-s + (0.361 + 2.97i)9-s + (0.5 − 0.866i)10-s + (0.402 − 0.697i)11-s + (−1.29 − 1.14i)12-s + (−2.61 − 1.51i)13-s + (2.97 − 1.71i)14-s + (0.548 + 1.64i)15-s + 16-s + (1.94 + 3.36i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.748 + 0.663i)3-s − 0.5·4-s + (0.387 + 0.223i)5-s + (0.468 − 0.529i)6-s + (0.649 + 1.12i)7-s + 0.353i·8-s + (0.120 + 0.992i)9-s + (0.158 − 0.273i)10-s + (0.121 − 0.210i)11-s + (−0.374 − 0.331i)12-s + (−0.725 − 0.418i)13-s + (0.794 − 0.458i)14-s + (0.141 + 0.424i)15-s + 0.250·16-s + (0.471 + 0.816i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.747 - 0.664i$
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: \((2,\ 930,\ (\ :1/2),\ 0.747 - 0.664i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96414 + 0.747300i\)
\(L(\frac12)\) \(\approx\) \(1.96414 + 0.747300i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.29 - 1.14i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (1.97 - 5.20i)T \)
good7 \( 1 + (-1.71 - 2.97i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.402 + 0.697i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.61 + 1.51i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.94 - 3.36i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.37 + 2.39i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.142T + 23T^{2} \)
29 \( 1 - 7.20T + 29T^{2} \)
37 \( 1 + (9.28 - 5.36i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.634 + 0.366i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.711 - 0.410i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 4.88iT - 47T^{2} \)
53 \( 1 + (-6.79 + 11.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-11.5 + 6.69i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 10.6iT - 61T^{2} \)
67 \( 1 + (0.342 - 0.593i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9.83 - 5.67i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.721 - 0.416i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.81 + 1.62i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.934 + 1.61i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.24T + 89T^{2} \)
97 \( 1 - 16.1T + 97T^{2} \)
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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

−10.17084172513358835860753682750, −9.437875035961330319822251233330, −8.471628663245961031332119675192, −8.248981026651404319768484039772, −6.76686984852111416260801228111, −5.36755242883130900712218045601, −4.91824649043200023360095859267, −3.58465045801001784034771801889, −2.67862471466453162323273597160, −1.82514764184438711924637030375, 0.966618175275109438066382802616, 2.27549486775257887111764235539, 3.78008994457003801087786459201, 4.64634982716835096269842887143, 5.74600039836430076640125026593, 6.97028484232261253899384548395, 7.28997250912212462802092294640, 8.156530863471595248371804472404, 8.943094556543489665305492004293, 9.798702762217200425388586942575

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