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Properties

Label 2-930-5.4-c1-0-0
Degree $2$
Conductor $930$
Sign $-0.200 - 0.979i$
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 + (0.447 + 2.19i)5-s − 6-s − 3.38i·7-s + i·8-s − 9-s + (2.19 − 0.447i)10-s − 6.27·11-s + i·12-s + 6.89i·13-s − 3.38·14-s + (2.19 − 0.447i)15-s + 16-s − 1.40i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (0.200 + 0.979i)5-s − 0.408·6-s − 1.27i·7-s + 0.353i·8-s − 0.333·9-s + (0.692 − 0.141i)10-s − 1.89·11-s + 0.288i·12-s + 1.91i·13-s − 0.903·14-s + (0.565 − 0.115i)15-s + 0.250·16-s − 0.341i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.200 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.200 - 0.979i$
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: \((2,\ 930,\ (\ :1/2),\ -0.200 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.104166 + 0.127598i\)
\(L(\frac12)\) \(\approx\) \(0.104166 + 0.127598i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.447 - 2.19i)T \)
31 \( 1 - T \)
good7 \( 1 + 3.38iT - 7T^{2} \)
11 \( 1 + 6.27T + 11T^{2} \)
13 \( 1 - 6.89iT - 13T^{2} \)
17 \( 1 + 1.40iT - 17T^{2} \)
19 \( 1 + 1.97T + 19T^{2} \)
23 \( 1 + 2.40iT - 23T^{2} \)
29 \( 1 + 7.79T + 29T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 9.58T + 41T^{2} \)
43 \( 1 - 3.38iT - 43T^{2} \)
47 \( 1 - 2.59iT - 47T^{2} \)
53 \( 1 - 8.40iT - 53T^{2} \)
59 \( 1 - 0.973T + 59T^{2} \)
61 \( 1 - 7.40T + 61T^{2} \)
67 \( 1 - 1.92iT - 67T^{2} \)
71 \( 1 - 0.540T + 71T^{2} \)
73 \( 1 + 1.43iT - 73T^{2} \)
79 \( 1 + 6.94T + 79T^{2} \)
83 \( 1 + 9.19iT - 83T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 + 10.6iT - 97T^{2} \)
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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

−10.39737996702642924786376461217, −9.803671088568408597423983109345, −8.661629216843652913407134814608, −7.54200778198801671637102103374, −7.12741207866359683834639781868, −6.13353763344838522930619084444, −4.85289405987524021816122538441, −3.85536816313664361228109378132, −2.71205513838039692944580665886, −1.80768634455546148930607158416, 0.07154066185754190947194523075, 2.32532413187893770101005317513, 3.52017977232265757727018159209, 5.10907662013859404269781812493, 5.31858402971488320460418665568, 5.93176213792738999685610177364, 7.59003750877674087885400425567, 8.332534476752644375068060409078, 8.690896900615333546563292121162, 9.841746554166867498580831334292

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