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Properties

Label 2-930-93.26-c1-0-28
Degree $2$
Conductor $930$
Sign $0.636 + 0.771i$
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 + (1.72 − 0.118i)3-s − 4-s + (0.866 − 0.5i)5-s + (−0.118 − 1.72i)6-s + (−0.777 + 1.34i)7-s + i·8-s + (2.97 − 0.409i)9-s + (−0.5 − 0.866i)10-s + (1.42 + 2.47i)11-s + (−1.72 + 0.118i)12-s + (−1.67 + 0.966i)13-s + (1.34 + 0.777i)14-s + (1.43 − 0.966i)15-s + 16-s + (2.05 − 3.55i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.997 − 0.0684i)3-s − 0.5·4-s + (0.387 − 0.223i)5-s + (−0.0484 − 0.705i)6-s + (−0.293 + 0.508i)7-s + 0.353i·8-s + (0.990 − 0.136i)9-s + (−0.158 − 0.273i)10-s + (0.430 + 0.745i)11-s + (−0.498 + 0.0342i)12-s + (−0.464 + 0.268i)13-s + (0.359 + 0.207i)14-s + (0.371 − 0.249i)15-s + 0.250·16-s + (0.497 − 0.862i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.636 + 0.771i$
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: \((2,\ 930,\ (\ :1/2),\ 0.636 + 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11827 - 0.998145i\)
\(L(\frac12)\) \(\approx\) \(2.11827 - 0.998145i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.118i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (-4.24 - 3.60i)T \)
good7 \( 1 + (0.777 - 1.34i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.42 - 2.47i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.67 - 0.966i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.05 + 3.55i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.49 + 4.32i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 8.12T + 23T^{2} \)
29 \( 1 + 1.76T + 29T^{2} \)
37 \( 1 + (2.22 + 1.28i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.09 - 0.629i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (9.09 + 5.25i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.59iT - 47T^{2} \)
53 \( 1 + (-5.37 - 9.30i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (11.5 + 6.65i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 4.90iT - 61T^{2} \)
67 \( 1 + (-0.173 - 0.300i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.0462 + 0.0266i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (14.2 - 8.20i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.174 + 0.100i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.60 + 2.77i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.22T + 89T^{2} \)
97 \( 1 - 0.201T + 97T^{2} \)
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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

−9.724207813610568060142490331460, −9.197199394289319338434091135385, −8.716702118248185173227629841248, −7.40072884823130479219790055233, −6.81137952580104168497277163069, −5.24309407403597606477292715144, −4.54234992797719480368948233410, −3.19749817028403734023515533926, −2.52962399801829589983657867656, −1.30769456067074089389508077304, 1.35092035423012096373066364219, 3.04636030153523427127884399799, 3.73423668187140071447694487249, 4.93861907886868155446843473510, 6.03617633483250537995163050758, 6.88714603890575066437808450445, 7.70752364688176751820996786327, 8.411602892336935960559809176598, 9.269769550000426994563943709048, 9.988420657349213422099042775975

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