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Properties

Label 2-930-93.26-c1-0-10
Degree $2$
Conductor $930$
Sign $-0.955 - 0.294i$
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 + (0.652 + 1.60i)3-s − 4-s + (−0.866 + 0.5i)5-s + (−1.60 + 0.652i)6-s + (2.20 − 3.81i)7-s − i·8-s + (−2.14 + 2.09i)9-s + (−0.5 − 0.866i)10-s + (2.82 + 4.88i)11-s + (−0.652 − 1.60i)12-s + (−1.84 + 1.06i)13-s + (3.81 + 2.20i)14-s + (−1.36 − 1.06i)15-s + 16-s + (−0.227 + 0.393i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.376 + 0.926i)3-s − 0.5·4-s + (−0.387 + 0.223i)5-s + (−0.654 + 0.266i)6-s + (0.832 − 1.44i)7-s − 0.353i·8-s + (−0.716 + 0.698i)9-s + (−0.158 − 0.273i)10-s + (0.850 + 1.47i)11-s + (−0.188 − 0.463i)12-s + (−0.510 + 0.294i)13-s + (1.01 + 0.588i)14-s + (−0.353 − 0.274i)15-s + 0.250·16-s + (−0.0550 + 0.0953i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.955 - 0.294i$
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: \((2,\ 930,\ (\ :1/2),\ -0.955 - 0.294i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.212984 + 1.41273i\)
\(L(\frac12)\) \(\approx\) \(0.212984 + 1.41273i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.652 - 1.60i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (2.37 - 5.03i)T \)
good7 \( 1 + (-2.20 + 3.81i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.82 - 4.88i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.84 - 1.06i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.227 - 0.393i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.13 - 7.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.55T + 23T^{2} \)
29 \( 1 - 0.364T + 29T^{2} \)
37 \( 1 + (-2.57 - 1.48i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.13 - 2.38i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.0 - 5.77i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.02iT - 47T^{2} \)
53 \( 1 + (-2.98 - 5.17i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.09 - 1.21i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 15.3iT - 61T^{2} \)
67 \( 1 + (1.30 + 2.25i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.565 + 0.326i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.70 - 1.56i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.77 - 4.49i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.53 + 14.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.28T + 89T^{2} \)
97 \( 1 - 14.1T + 97T^{2} \)
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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

−10.22730869155975692449772287608, −9.732695281191490727008768268611, −8.661204491143882307162773356552, −7.79293229878784779830412803399, −7.29880752358006137673356631504, −6.27375787335138586184510933756, −4.82259731038147613937245320813, −4.31467170886745526571210555488, −3.69467954342666091125455521006, −1.78714572161963876106355290327, 0.65357609887054911947509933230, 2.11829078428468371416423484792, 2.84582175831717957847217836675, 4.11341053472278057102631506146, 5.37999461915626729193188700458, 6.10856633160548509355442276822, 7.31353988150868691984699863180, 8.403077609377310300595493116004, 8.732467490065192119672366103641, 9.321754551369027805327111745131

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