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Properties

Label 2-930-5.4-c1-0-3
Degree $2$
Conductor $930$
Sign $-0.894 - 0.447i$
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 + i)5-s + 6-s + 2i·7-s − i·8-s − 9-s + (−1 + 2i)10-s − 6·11-s + i·12-s + 2i·13-s − 2·14-s + (1 − 2i)15-s + 16-s + 6i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (0.894 + 0.447i)5-s + 0.408·6-s + 0.755i·7-s − 0.353i·8-s − 0.333·9-s + (−0.316 + 0.632i)10-s − 1.80·11-s + 0.288i·12-s + 0.554i·13-s − 0.534·14-s + (0.258 − 0.516i)15-s + 0.250·16-s + 1.45i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.894 - 0.447i$
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: \((2,\ 930,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.222207 + 0.941287i\)
\(L(\frac12)\) \(\approx\) \(0.222207 + 0.941287i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - i)T \)
31 \( 1 + T \)
good7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 + 8T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 10iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 16T + 71T^{2} \)
73 \( 1 - 8iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 16iT - 97T^{2} \)
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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

−10.51562922229609008700824318812, −9.458820960308478406152581506333, −8.423286872269524714293322793995, −8.082992993984181268186689906233, −6.78361127894603513312779486550, −6.21364179357907683663796927527, −5.53786055272347795924294184933, −4.47727261522209926947945209115, −2.77623432088683007099835644420, −1.97375645400248099072466309148, 0.41938997965137985629949346235, 2.19071424827379465570161767971, 3.07971093944410303561855635872, 4.45309952440051840068136497396, 5.09057196258572380238194577904, 5.92531900103016535393686587314, 7.32187816876209936881109139987, 8.252289531777121079229109701580, 9.114882570523260308065727861260, 9.920692181020961282739520091759

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