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Properties

Label 2-930-93.26-c1-0-4
Degree $2$
Conductor $930$
Sign $0.791 - 0.611i$
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 + (−1.69 − 0.345i)3-s − 4-s + (−0.866 + 0.5i)5-s + (−0.345 + 1.69i)6-s + (0.742 − 1.28i)7-s + i·8-s + (2.76 + 1.17i)9-s + (0.5 + 0.866i)10-s + (−2.14 − 3.71i)11-s + (1.69 + 0.345i)12-s + (−1.46 + 0.848i)13-s + (−1.28 − 0.742i)14-s + (1.64 − 0.549i)15-s + 16-s + (−2.87 + 4.97i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.979 − 0.199i)3-s − 0.5·4-s + (−0.387 + 0.223i)5-s + (−0.141 + 0.692i)6-s + (0.280 − 0.486i)7-s + 0.353i·8-s + (0.920 + 0.391i)9-s + (0.158 + 0.273i)10-s + (−0.645 − 1.11i)11-s + (0.489 + 0.0998i)12-s + (−0.407 + 0.235i)13-s + (−0.343 − 0.198i)14-s + (0.424 − 0.141i)15-s + 0.250·16-s + (−0.697 + 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.791 - 0.611i$
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: \((2,\ 930,\ (\ :1/2),\ 0.791 - 0.611i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.501645 + 0.171331i\)
\(L(\frac12)\) \(\approx\) \(0.501645 + 0.171331i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 + 0.345i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (-2.50 + 4.97i)T \)
good7 \( 1 + (-0.742 + 1.28i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.14 + 3.71i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.46 - 0.848i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.87 - 4.97i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.59 - 2.77i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.16T + 23T^{2} \)
29 \( 1 - 7.18T + 29T^{2} \)
37 \( 1 + (-8.23 - 4.75i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (9.76 - 5.63i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-6.44 - 3.72i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 11.9iT - 47T^{2} \)
53 \( 1 + (-3.03 - 5.25i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.18 - 0.682i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 12.5iT - 61T^{2} \)
67 \( 1 + (-5.98 - 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.650 - 0.375i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.41 - 1.96i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-9.64 - 5.56i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.88 + 8.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 4.38T + 89T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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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

−10.47095557661673271274239593789, −9.693825945942707753852498027719, −8.205412806172868587192913049005, −7.919055150757200708554240434917, −6.49742155412304096323144162582, −5.91092171741729750200600497230, −4.62783153471550613193648546281, −4.02787583875104451861235375013, −2.59957902501973378646476715518, −1.14450758286537946278794544404, 0.32814493080095603466527817990, 2.35718657482055459505273970231, 4.15595394726022437623593429082, 4.93923758479717170952389705794, 5.40902114556695773460532351405, 6.72306765779545532955522107126, 7.17981382997464171889678763942, 8.207340712201884300271734198760, 9.120260939425282498642812004972, 10.01054618130853053145473064857

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