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Properties

Label 2-930-31.2-c1-0-10
Degree $2$
Conductor $930$
Sign $0.321 - 0.946i$
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  + (0.809 + 0.587i)2-s + (0.809 − 0.587i)3-s + (0.309 + 0.951i)4-s + 5-s + 6-s + (0.782 + 2.40i)7-s + (−0.309 + 0.951i)8-s + (0.309 − 0.951i)9-s + (0.809 + 0.587i)10-s + (1.74 + 5.36i)11-s + (0.809 + 0.587i)12-s + (−2.75 + 2.00i)13-s + (−0.782 + 2.40i)14-s + (0.809 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (0.425 − 1.30i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.467 − 0.339i)3-s + (0.154 + 0.475i)4-s + 0.447·5-s + 0.408·6-s + (0.295 + 0.909i)7-s + (−0.109 + 0.336i)8-s + (0.103 − 0.317i)9-s + (0.255 + 0.185i)10-s + (0.525 + 1.61i)11-s + (0.233 + 0.169i)12-s + (−0.763 + 0.555i)13-s + (−0.209 + 0.643i)14-s + (0.208 − 0.151i)15-s + (−0.202 + 0.146i)16-s + (0.103 − 0.317i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.321 - 0.946i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.321 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21737 + 1.58912i\)
\(L(\frac12)\) \(\approx\) \(2.21737 + 1.58912i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 - T \)
31 \( 1 + (-4.49 + 3.28i)T \)
good7 \( 1 + (-0.782 - 2.40i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-1.74 - 5.36i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (2.75 - 2.00i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.425 + 1.30i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.49 + 2.53i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (0.113 - 0.350i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (3.58 + 2.60i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 - 6.36T + 37T^{2} \)
41 \( 1 + (-0.624 - 0.454i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-5.77 - 4.19i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (3.56 - 2.59i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.96 + 9.12i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-7.11 + 5.16i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 - 5.92T + 61T^{2} \)
67 \( 1 + 7.12T + 67T^{2} \)
71 \( 1 + (1.23 - 3.81i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.80 + 5.55i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (1.19 - 3.67i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.0617 + 0.0448i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (4.92 + 15.1i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-4.73 - 14.5i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.863267120729541274912345084971, −9.412211016299218916010738232358, −8.505801639623743086953788695185, −7.52432664905042449638871157285, −6.83542020659271218724799966919, −5.99635903312937522889135838264, −4.87878990578940309602105851293, −4.20775022319450854335323195294, −2.56544430030832973131095877138, −1.99653682467151888135754798472, 1.07776673527835734863022456510, 2.54658460396866290424717269135, 3.57883179661153735399987701372, 4.33045842111258864200180796153, 5.46906181616238708231779755869, 6.24702981403461727227228563898, 7.37997593657011615635784991762, 8.340778776218228981154146508344, 9.126798065312072124656399573983, 10.20969579397428193367621294926

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