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Properties

Label 2-930-93.68-c1-0-34
Degree $2$
Conductor $930$
Sign $-0.691 + 0.722i$
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 + (−1.20 − 1.24i)3-s − 4-s + (0.866 + 0.5i)5-s + (1.24 − 1.20i)6-s + (−1.64 − 2.85i)7-s − i·8-s + (−0.0972 + 2.99i)9-s + (−0.5 + 0.866i)10-s + (0.264 − 0.458i)11-s + (1.20 + 1.24i)12-s + (2.90 + 1.68i)13-s + (2.85 − 1.64i)14-s + (−0.421 − 1.68i)15-s + 16-s + (−2.88 − 5.00i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.695 − 0.718i)3-s − 0.5·4-s + (0.387 + 0.223i)5-s + (0.508 − 0.491i)6-s + (−0.622 − 1.07i)7-s − 0.353i·8-s + (−0.0324 + 0.999i)9-s + (−0.158 + 0.273i)10-s + (0.0798 − 0.138i)11-s + (0.347 + 0.359i)12-s + (0.807 + 0.465i)13-s + (0.762 − 0.440i)14-s + (−0.108 − 0.433i)15-s + 0.250·16-s + (−0.700 − 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.691 + 0.722i$
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: \((2,\ 930,\ (\ :1/2),\ -0.691 + 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.156830 - 0.367482i\)
\(L(\frac12)\) \(\approx\) \(0.156830 - 0.367482i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.20 + 1.24i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (-1.14 + 5.44i)T \)
good7 \( 1 + (1.64 + 2.85i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.264 + 0.458i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.90 - 1.68i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.88 + 5.00i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.83 - 3.17i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 8.32T + 23T^{2} \)
29 \( 1 + 4.28T + 29T^{2} \)
37 \( 1 + (4.77 - 2.75i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.40 + 3.69i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.37 - 3.10i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.4iT - 47T^{2} \)
53 \( 1 + (3.33 - 5.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.57 - 3.79i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 10.5iT - 61T^{2} \)
67 \( 1 + (-7.82 + 13.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (12.0 + 6.97i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.32 + 3.64i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.56 - 4.36i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.38 + 9.32i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 15.0T + 89T^{2} \)
97 \( 1 + 14.7T + 97T^{2} \)
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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

−9.882595222627034180837120902961, −8.796228096921943074904724990661, −7.72733178901096339503612077464, −7.09638370707639705186753155050, −6.36219738633541612666519496038, −5.76766204153589000202241942446, −4.57830543632200839802122384015, −3.51829859386588039829329847347, −1.76726372123032617382512753323, −0.20177183550806808199022981349, 1.73474218574912784528096754894, 3.12672490435426364292891591963, 4.03519448242362738794876673921, 5.15758723929767094592326770085, 5.90440801077434159454721338420, 6.55936323490048131076991942174, 8.320255844073892740955622661589, 8.945193163018498138170167656731, 9.708978791950349669291284993395, 10.34940640873878796530126733265

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