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Properties

Label 2-930-93.68-c1-0-23
Degree $2$
Conductor $930$
Sign $-0.521 - 0.853i$
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 + (1.71 + 0.237i)3-s − 4-s + (0.866 + 0.5i)5-s + (−0.237 + 1.71i)6-s + (2.20 + 3.81i)7-s − i·8-s + (2.88 + 0.813i)9-s + (−0.5 + 0.866i)10-s + (−2.82 + 4.88i)11-s + (−1.71 − 0.237i)12-s + (−1.84 − 1.06i)13-s + (−3.81 + 2.20i)14-s + (1.36 + 1.06i)15-s + 16-s + (0.227 + 0.393i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.990 + 0.136i)3-s − 0.5·4-s + (0.387 + 0.223i)5-s + (−0.0967 + 0.700i)6-s + (0.832 + 1.44i)7-s − 0.353i·8-s + (0.962 + 0.271i)9-s + (−0.158 + 0.273i)10-s + (−0.850 + 1.47i)11-s + (−0.495 − 0.0684i)12-s + (−0.510 − 0.294i)13-s + (−1.01 + 0.588i)14-s + (0.353 + 0.274i)15-s + 0.250·16-s + (0.0550 + 0.0953i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.521 - 0.853i$
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: \((2,\ 930,\ (\ :1/2),\ -0.521 - 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10926 + 1.97730i\)
\(L(\frac12)\) \(\approx\) \(1.10926 + 1.97730i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 0.237i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (2.37 + 5.03i)T \)
good7 \( 1 + (-2.20 - 3.81i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.82 - 4.88i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.84 + 1.06i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.227 - 0.393i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.13 + 7.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.55T + 23T^{2} \)
29 \( 1 + 0.364T + 29T^{2} \)
37 \( 1 + (-2.57 + 1.48i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.13 - 2.38i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-10.0 + 5.77i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.02iT - 47T^{2} \)
53 \( 1 + (2.98 - 5.17i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.09 - 1.21i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 15.3iT - 61T^{2} \)
67 \( 1 + (1.30 - 2.25i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.565 + 0.326i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.70 + 1.56i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.77 + 4.49i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.53 + 14.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.28T + 89T^{2} \)
97 \( 1 - 14.1T + 97T^{2} \)
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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

−10.08308986182244107963272568093, −9.128851851698227252149251462953, −8.855062277493375398514015370886, −7.67016341748391744991302634265, −7.32663679468418491902992330221, −6.01619594683528409298526745113, −4.99560510555949174613119038188, −4.45016520184121934667186390709, −2.60790954027942113854060167592, −2.17580345294509663236446468679, 0.997449322303694510330254618704, 2.09935898844319839086748174979, 3.33292216637527438585190948036, 4.13959884765459652510933679435, 5.10279428635150537634781756036, 6.40192801902421624149982693911, 7.76523113702049224828516270789, 8.017879557450445863604722729747, 8.974967625605186069097772330266, 9.883303608961169542713976085517

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