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Properties

Label 2-930-465.254-c1-0-10
Degree $2$
Conductor $930$
Sign $0.0603 - 0.998i$
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  − 2-s + (−1.66 + 0.469i)3-s + 4-s + (−1.78 + 1.35i)5-s + (1.66 − 0.469i)6-s + (−2.19 + 1.26i)7-s − 8-s + (2.55 − 1.56i)9-s + (1.78 − 1.35i)10-s + (0.583 − 1.01i)11-s + (−1.66 + 0.469i)12-s + (3.18 − 5.51i)13-s + (2.19 − 1.26i)14-s + (2.33 − 3.09i)15-s + 16-s + (3.41 − 1.97i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.962 + 0.271i)3-s + 0.5·4-s + (−0.796 + 0.604i)5-s + (0.680 − 0.191i)6-s + (−0.828 + 0.478i)7-s − 0.353·8-s + (0.852 − 0.522i)9-s + (0.563 − 0.427i)10-s + (0.175 − 0.304i)11-s + (−0.481 + 0.135i)12-s + (0.882 − 1.52i)13-s + (0.585 − 0.338i)14-s + (0.602 − 0.798i)15-s + 0.250·16-s + (0.828 − 0.478i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0603 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0603 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.0603 - 0.998i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.0603 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.371481 + 0.349683i\)
\(L(\frac12)\) \(\approx\) \(0.371481 + 0.349683i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (1.66 - 0.469i)T \)
5 \( 1 + (1.78 - 1.35i)T \)
31 \( 1 + (1.00 - 5.47i)T \)
good7 \( 1 + (2.19 - 1.26i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.583 + 1.01i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.18 + 5.51i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.41 + 1.97i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.33 - 2.31i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.97iT - 23T^{2} \)
29 \( 1 + 4.97T + 29T^{2} \)
37 \( 1 + (-2.60 - 4.50i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.504 - 0.291i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.106 - 0.184i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.60T + 47T^{2} \)
53 \( 1 + (5.05 + 2.92i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-10.0 + 5.78i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 7.81iT - 61T^{2} \)
67 \( 1 + (-2.19 - 1.26i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-13.2 - 7.65i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.80 - 6.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.30 - 5.37i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.79 - 1.61i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 + 11.6iT - 97T^{2} \)
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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

−10.15497771188399325756870249910, −9.767634594068819825933944710149, −8.563050921861814387924740086574, −7.72935531240043938833676240547, −6.88544575618134446772637530392, −5.99932910150604942620991226392, −5.36454473975965371492596851821, −3.66581257321445153511804216211, −3.10375554829546148650534760596, −0.953171933234104402395032649825, 0.47518604372205768550915756469, 1.70986669958431239234278351785, 3.67278907421833296124152977471, 4.44361914711428377117854333307, 5.74566608789793735718163988389, 6.62949866832806081338766678042, 7.25723064987542367590108660314, 8.110156299050752890009721632742, 9.194197471312206485851145028419, 9.750782486832764556128735826836

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