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Properties

Label 2-930-5.4-c1-0-1
Degree $2$
Conductor $930$
Sign $-0.447 + 0.894i$
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 + (−1 + 2i)5-s − 6-s + i·7-s − i·8-s − 9-s + (−2 − i)10-s − 3·11-s − i·12-s + 4i·13-s − 14-s + (−2 − i)15-s + 16-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.447 + 0.894i)5-s − 0.408·6-s + 0.377i·7-s − 0.353i·8-s − 0.333·9-s + (−0.632 − 0.316i)10-s − 0.904·11-s − 0.288i·12-s + 1.10i·13-s − 0.267·14-s + (−0.516 − 0.258i)15-s + 0.250·16-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.305556 - 0.494401i\)
\(L(\frac12)\) \(\approx\) \(0.305556 - 0.494401i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 2i)T \)
31 \( 1 + T \)
good7 \( 1 - iT - 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + 5iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 5iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + 5iT - 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 + 5T + 71T^{2} \)
73 \( 1 - 7iT - 73T^{2} \)
79 \( 1 + 3T + 79T^{2} \)
83 \( 1 - 2iT - 83T^{2} \)
89 \( 1 + T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−10.44008843758896986894137273194, −9.860805618875227593481910657206, −8.800218478491754257952755942119, −8.154227292792793092578435497507, −7.14030689535698297919481576656, −6.48148449414858039901327596220, −5.43887489988290248425580158710, −4.52910691332778202043452752015, −3.54288568469892368614829201748, −2.40531935874045117295226274603, 0.27446055720847916293724345868, 1.51866075989609014854314795593, 2.92225776139589165403907765123, 3.92168080914936600301426973713, 5.12899848982183335966401969341, 5.69711276393135988105786699428, 7.30237574232669532135802547708, 7.86416968838300278318250902741, 8.627607386761118794623801111332, 9.545010160273345635842361833700

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