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Properties

Label 2-930-155.129-c1-0-11
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: $\chi_{930} (439, \cdot )$
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

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

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