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Properties

Label 2-930-465.464-c1-0-9
Degree $2$
Conductor $930$
Sign $-0.0663 - 0.997i$
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 + (−1.36 + 1.06i)3-s + 4-s + (−2.16 − 0.546i)5-s + (1.36 − 1.06i)6-s + 1.97i·7-s − 8-s + (0.733 − 2.90i)9-s + (2.16 + 0.546i)10-s + 0.980·11-s + (−1.36 + 1.06i)12-s + 4.39·13-s − 1.97i·14-s + (3.54 − 1.56i)15-s + 16-s − 2.02i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.788 + 0.614i)3-s + 0.5·4-s + (−0.969 − 0.244i)5-s + (0.557 − 0.434i)6-s + 0.746i·7-s − 0.353·8-s + (0.244 − 0.969i)9-s + (0.685 + 0.172i)10-s + 0.295·11-s + (−0.394 + 0.307i)12-s + 1.21·13-s − 0.527i·14-s + (0.915 − 0.403i)15-s + 0.250·16-s − 0.490i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0663 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0663 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.0663 - 0.997i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.0663 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.398625 + 0.425993i\)
\(L(\frac12)\) \(\approx\) \(0.398625 + 0.425993i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.36 - 1.06i)T \)
5 \( 1 + (2.16 + 0.546i)T \)
31 \( 1 + (1.90 - 5.23i)T \)
good7 \( 1 - 1.97iT - 7T^{2} \)
11 \( 1 - 0.980T + 11T^{2} \)
13 \( 1 - 4.39T + 13T^{2} \)
17 \( 1 + 2.02iT - 17T^{2} \)
19 \( 1 + 3.72T + 19T^{2} \)
23 \( 1 + 6.09iT - 23T^{2} \)
29 \( 1 - 2.54T + 29T^{2} \)
37 \( 1 - 9.44T + 37T^{2} \)
41 \( 1 - 12.5iT - 41T^{2} \)
43 \( 1 + 3.33T + 43T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 - 7.67iT - 53T^{2} \)
59 \( 1 - 0.348iT - 59T^{2} \)
61 \( 1 + 5.15iT - 61T^{2} \)
67 \( 1 - 1.03iT - 67T^{2} \)
71 \( 1 - 8.70iT - 71T^{2} \)
73 \( 1 + 5.88T + 73T^{2} \)
79 \( 1 - 8.91iT - 79T^{2} \)
83 \( 1 - 8.99iT - 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 - 17.1iT - 97T^{2} \)
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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

−10.36954843570917165071775100254, −9.366467379738813687607398832604, −8.660250143990488038394346501512, −8.073064298809279332971365448398, −6.69333674414235598645074302517, −6.20514782278825594586273419822, −4.97010735242695907098259793670, −4.10780122274449709558337907260, −2.95384857537782837708740431847, −1.04585246050435497911934536798, 0.49742715054215699758001434175, 1.75072066394562592794373113221, 3.50028022944928088729561541577, 4.39583598672424622806379028931, 5.85883059571257551297829865150, 6.58687989713593747554729654307, 7.39417587919309159906236924893, 8.011356576084153825325705823408, 8.815307529096153555273168849315, 10.09977631385417892938143594511

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