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Properties

Label 2-930-5.4-c1-0-4
Degree $2$
Conductor $930$
Sign $0.959 - 0.280i$
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 − i·3-s − 4-s + (−2.14 + 0.627i)5-s − 6-s − 0.255i·7-s + i·8-s − 9-s + (0.627 + 2.14i)10-s + 2.03·11-s + i·12-s + 1.70i·13-s − 0.255·14-s + (0.627 + 2.14i)15-s + 16-s + 5.83i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.959 + 0.280i)5-s − 0.408·6-s − 0.0964i·7-s + 0.353i·8-s − 0.333·9-s + (0.198 + 0.678i)10-s + 0.614·11-s + 0.288i·12-s + 0.473i·13-s − 0.0681·14-s + (0.162 + 0.554i)15-s + 0.250·16-s + 1.41i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.280i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 - 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.959 - 0.280i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.959 - 0.280i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.896119 + 0.128326i\)
\(L(\frac12)\) \(\approx\) \(0.896119 + 0.128326i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 + (2.14 - 0.627i)T \)
31 \( 1 - T \)
good7 \( 1 + 0.255iT - 7T^{2} \)
11 \( 1 - 2.03T + 11T^{2} \)
13 \( 1 - 1.70iT - 13T^{2} \)
17 \( 1 - 5.83iT - 17T^{2} \)
19 \( 1 + 6.09T + 19T^{2} \)
23 \( 1 - 4.83iT - 23T^{2} \)
29 \( 1 - 2.58T + 29T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 11.1T + 41T^{2} \)
43 \( 1 - 0.255iT - 43T^{2} \)
47 \( 1 - 9.83iT - 47T^{2} \)
53 \( 1 - 1.16iT - 53T^{2} \)
59 \( 1 - 5.09T + 59T^{2} \)
61 \( 1 - 0.160T + 61T^{2} \)
67 \( 1 + 7.38iT - 67T^{2} \)
71 \( 1 + 5.64T + 71T^{2} \)
73 \( 1 - 9.93iT - 73T^{2} \)
79 \( 1 + 15.1T + 79T^{2} \)
83 \( 1 - 8.42iT - 83T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 - 4.87iT - 97T^{2} \)
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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

−10.36019751932403737769035405333, −9.191139221037191315633892515740, −8.444677496671700801275640934962, −7.71017603000176916106542868770, −6.72957701907450987536875014463, −5.90795553461681228344939407726, −4.34704890597374620454112435075, −3.83717696812735753892781644990, −2.55562327694977164531846327964, −1.28640420678961987446190023845, 0.47980034374181565795987252548, 2.79782609462110289642465101269, 4.07560655381895995282978745950, 4.59957708624467858620712586220, 5.63967621378665196005227340396, 6.70628609147252988705699630840, 7.47829180347776677973065330056, 8.537164115714719925499328096909, 8.860520925298505985271384666271, 9.930563479378253778052293690262

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