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Properties

Label 2-930-155.149-c1-0-28
Degree $2$
Conductor $930$
Sign $0.630 + 0.776i$
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 + (0.866 − 0.5i)3-s − 4-s + (−0.347 − 2.20i)5-s + (0.5 + 0.866i)6-s + (3.10 − 1.79i)7-s − i·8-s + (0.499 − 0.866i)9-s + (2.20 − 0.347i)10-s + (0.121 − 0.209i)11-s + (−0.866 + 0.5i)12-s + (0.613 + 0.354i)13-s + (1.79 + 3.10i)14-s + (−1.40 − 1.73i)15-s + 16-s + (−3.14 + 1.81i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.499 − 0.288i)3-s − 0.5·4-s + (−0.155 − 0.987i)5-s + (0.204 + 0.353i)6-s + (1.17 − 0.678i)7-s − 0.353i·8-s + (0.166 − 0.288i)9-s + (0.698 − 0.109i)10-s + (0.0364 − 0.0631i)11-s + (−0.249 + 0.144i)12-s + (0.170 + 0.0982i)13-s + (0.479 + 0.830i)14-s + (−0.362 − 0.449i)15-s + 0.250·16-s + (−0.763 + 0.440i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 + 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.630 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.630 + 0.776i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.630 + 0.776i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60160 - 0.762199i\)
\(L(\frac12)\) \(\approx\) \(1.60160 - 0.762199i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (0.347 + 2.20i)T \)
31 \( 1 + (-2.15 + 5.13i)T \)
good7 \( 1 + (-3.10 + 1.79i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.121 + 0.209i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.613 - 0.354i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.14 - 1.81i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.13 + 7.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.53iT - 23T^{2} \)
29 \( 1 + 0.496T + 29T^{2} \)
37 \( 1 + (-1.31 + 0.757i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.25 + 5.63i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.87 + 1.65i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.95iT - 47T^{2} \)
53 \( 1 + (-3.26 - 1.88i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.36 + 4.09i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 12.8T + 61T^{2} \)
67 \( 1 + (-7.72 - 4.46i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.20 - 5.55i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.51 - 2.60i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.44 + 4.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.82 - 2.78i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 + 0.598iT - 97T^{2} \)
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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

−9.596029839749765210517396150892, −8.790890868513273236942768258136, −8.304249356596100087988732112714, −7.53355596424367215779580542387, −6.72371265548711782806708561875, −5.55501419466819503560869862173, −4.50375852599762964258807185879, −4.09215488426742796550218350143, −2.18006978887891577580965303634, −0.819014265507271421741234611910, 1.82896097351469611671280325320, 2.62310828859764810518165408698, 3.75838881324276625441297512517, 4.62404375997125308972058516626, 5.72699948551693410131311777534, 6.83834898516232368199399784529, 8.086611672002038653229949276368, 8.398324304088235592038763526184, 9.470413011461434635693386993577, 10.35915236571395271080694171442

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