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Properties

Label 2-930-93.92-c1-0-20
Degree $2$
Conductor $930$
Sign $0.998 + 0.0630i$
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 + (1.36 + 1.06i)3-s − 4-s − i·5-s + (1.06 − 1.36i)6-s + 1.28·7-s + i·8-s + (0.744 + 2.90i)9-s − 10-s + 0.392·11-s + (−1.36 − 1.06i)12-s + 4.84i·13-s − 1.28i·14-s + (1.06 − 1.36i)15-s + 16-s + 6.52·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.789 + 0.613i)3-s − 0.5·4-s − 0.447i·5-s + (0.433 − 0.558i)6-s + 0.484·7-s + 0.353i·8-s + (0.248 + 0.968i)9-s − 0.316·10-s + 0.118·11-s + (−0.394 − 0.306i)12-s + 1.34i·13-s − 0.342i·14-s + (0.274 − 0.353i)15-s + 0.250·16-s + 1.58·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.998 + 0.0630i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.998 + 0.0630i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12512 - 0.0671101i\)
\(L(\frac12)\) \(\approx\) \(2.12512 - 0.0671101i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.36 - 1.06i)T \)
5 \( 1 + iT \)
31 \( 1 + (-4.60 + 3.12i)T \)
good7 \( 1 - 1.28T + 7T^{2} \)
11 \( 1 - 0.392T + 11T^{2} \)
13 \( 1 - 4.84iT - 13T^{2} \)
17 \( 1 - 6.52T + 17T^{2} \)
19 \( 1 + 4.77T + 19T^{2} \)
23 \( 1 - 2.51T + 23T^{2} \)
29 \( 1 - 4.57T + 29T^{2} \)
37 \( 1 + 2.12iT - 37T^{2} \)
41 \( 1 - 5.07iT - 41T^{2} \)
43 \( 1 + 3.28iT - 43T^{2} \)
47 \( 1 + 0.834iT - 47T^{2} \)
53 \( 1 - 0.123T + 53T^{2} \)
59 \( 1 + 5.86iT - 59T^{2} \)
61 \( 1 + 9.87iT - 61T^{2} \)
67 \( 1 - 10.9T + 67T^{2} \)
71 \( 1 - 9.33iT - 71T^{2} \)
73 \( 1 - 9.18iT - 73T^{2} \)
79 \( 1 - 6.12iT - 79T^{2} \)
83 \( 1 - 2.73T + 83T^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 + 17.9T + 97T^{2} \)
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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

−9.872860136932223890694637413725, −9.427812872504415446024186934008, −8.438868789120000044486733721588, −8.029434625398504920846011026177, −6.70459720049828334151887295946, −5.29545879124177650840197105410, −4.49213538414078159879006869450, −3.76265913497905727530242971115, −2.54888188947719302610483510156, −1.45546909627345009869736826255, 1.10192895873830750144237681807, 2.71943531489524008864773601692, 3.60127938932023106521610037962, 4.91023767512623663158928141954, 5.97552550192850749754743503118, 6.76410882562035802626756376442, 7.74981512485053276195332610260, 8.104907267278575979781018883160, 8.945547228583078510807149619676, 10.02606393559860207880866230393

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