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Properties

Label 2-930-5.4-c3-0-54
Degree $2$
Conductor $930$
Sign $0.953 - 0.302i$
Analytic cond. $54.8717$
Root an. cond. $7.40754$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 3i·3-s − 4·4-s + (10.6 − 3.37i)5-s + 6·6-s + 22.4i·7-s − 8i·8-s − 9·9-s + (6.75 + 21.3i)10-s − 57.5·11-s + 12i·12-s − 41.6i·13-s − 44.8·14-s + (−10.1 − 31.9i)15-s + 16·16-s − 41.4i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (0.953 − 0.302i)5-s + 0.408·6-s + 1.21i·7-s − 0.353i·8-s − 0.333·9-s + (0.213 + 0.674i)10-s − 1.57·11-s + 0.288i·12-s − 0.888i·13-s − 0.856·14-s + (−0.174 − 0.550i)15-s + 0.250·16-s − 0.592i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.953 - 0.302i$
Analytic conductor: \(54.8717\)
Root analytic conductor: \(7.40754\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :3/2),\ 0.953 - 0.302i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.053281502\)
\(L(\frac12)\) \(\approx\) \(2.053281502\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
3 \( 1 + 3iT \)
5 \( 1 + (-10.6 + 3.37i)T \)
31 \( 1 + 31T \)
good7 \( 1 - 22.4iT - 343T^{2} \)
11 \( 1 + 57.5T + 1.33e3T^{2} \)
13 \( 1 + 41.6iT - 2.19e3T^{2} \)
17 \( 1 + 41.4iT - 4.91e3T^{2} \)
19 \( 1 - 55.8T + 6.85e3T^{2} \)
23 \( 1 - 115. iT - 1.21e4T^{2} \)
29 \( 1 - 220.T + 2.43e4T^{2} \)
37 \( 1 + 40.0iT - 5.06e4T^{2} \)
41 \( 1 - 236.T + 6.89e4T^{2} \)
43 \( 1 + 52.3iT - 7.95e4T^{2} \)
47 \( 1 - 3.59iT - 1.03e5T^{2} \)
53 \( 1 + 263. iT - 1.48e5T^{2} \)
59 \( 1 - 212.T + 2.05e5T^{2} \)
61 \( 1 - 678.T + 2.26e5T^{2} \)
67 \( 1 + 736. iT - 3.00e5T^{2} \)
71 \( 1 + 379.T + 3.57e5T^{2} \)
73 \( 1 - 1.07e3iT - 3.89e5T^{2} \)
79 \( 1 - 288.T + 4.93e5T^{2} \)
83 \( 1 - 876. iT - 5.71e5T^{2} \)
89 \( 1 - 1.11e3T + 7.04e5T^{2} \)
97 \( 1 + 1.30e3iT - 9.12e5T^{2} \)
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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

−9.605109137681080725990138759377, −8.701614542615292538862897814646, −8.048467711939308819505929662893, −7.22825281381832117152466683620, −6.09905788060374680850822783895, −5.40836594223610292302561317452, −5.08117451727466250221971150754, −3.01945280630030014647187305694, −2.25925412344018846733872599229, −0.73152887504090409847633838249, 0.815613400531191951118960805537, 2.20956536764068387407805032028, 3.10904421694600846816162735342, 4.28272186657817375522381601952, 5.00534299398775249712446479483, 6.06601471930519857292729678233, 7.10348154618082265550289729992, 8.115405030707086097774357765688, 9.080425232961450001426572795863, 9.993575412009408389549652540007

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