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Properties

Label 2-930-93.68-c1-0-27
Degree $2$
Conductor $930$
Sign $0.785 + 0.619i$
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 + (1.55 + 0.761i)3-s − 4-s + (−0.866 − 0.5i)5-s + (0.761 − 1.55i)6-s + (0.295 + 0.511i)7-s + i·8-s + (1.83 + 2.36i)9-s + (−0.5 + 0.866i)10-s + (2.33 − 4.05i)11-s + (−1.55 − 0.761i)12-s + (2.48 + 1.43i)13-s + (0.511 − 0.295i)14-s + (−0.966 − 1.43i)15-s + 16-s + (−1.50 − 2.60i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.898 + 0.439i)3-s − 0.5·4-s + (−0.387 − 0.223i)5-s + (0.310 − 0.635i)6-s + (0.111 + 0.193i)7-s + 0.353i·8-s + (0.613 + 0.789i)9-s + (−0.158 + 0.273i)10-s + (0.705 − 1.22i)11-s + (−0.449 − 0.219i)12-s + (0.690 + 0.398i)13-s + (0.136 − 0.0788i)14-s + (−0.249 − 0.371i)15-s + 0.250·16-s + (−0.364 − 0.631i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 + 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.785 + 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.785 + 0.619i$
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: \((2,\ 930,\ (\ :1/2),\ 0.785 + 0.619i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98302 - 0.688042i\)
\(L(\frac12)\) \(\approx\) \(1.98302 - 0.688042i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.55 - 0.761i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 + (-4.99 - 2.45i)T \)
good7 \( 1 + (-0.295 - 0.511i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.33 + 4.05i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.48 - 1.43i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.50 + 2.60i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.44 - 2.49i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.585T + 23T^{2} \)
29 \( 1 - 5.47T + 29T^{2} \)
37 \( 1 + (-4.88 + 2.81i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.52 + 2.61i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.86 + 1.65i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.89iT - 47T^{2} \)
53 \( 1 + (0.900 - 1.56i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.70 - 1.56i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 3.09iT - 61T^{2} \)
67 \( 1 + (5.77 - 9.99i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.46 - 0.844i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.55 - 1.47i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.835 + 0.482i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.93 + 10.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 0.409T + 89T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−9.979959073009144870153951487406, −8.928531480130900448414601759606, −8.698409472158596392435893772261, −7.82353450890329863128446947770, −6.58250389131387604558049338155, −5.35978414223384592548736584037, −4.25461575978055692228930251754, −3.58481527520847244226003028909, −2.61975450180455424680548586708, −1.18681814153647084969478021112, 1.28326743218193092122395077107, 2.78963300555857776357830737214, 3.97962040562346100747150857226, 4.66086585553466102025213333761, 6.29028737115464306218296086414, 6.76726382132143745267827810668, 7.75072856715154945648850587540, 8.215664252927607869927128329172, 9.193626500705960259842819351068, 9.834342547391298329539507476725

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