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Properties

Label 2-930-5.4-c1-0-22
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 + 0.828i·7-s + i·8-s − 9-s + (−1 − 2i)10-s − 0.828·11-s + i·12-s − 4.82i·13-s + 0.828·14-s + (−1 − 2i)15-s + 16-s + 0.828i·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 + 0.313i·7-s + 0.353i·8-s − 0.333·9-s + (−0.316 − 0.632i)10-s − 0.249·11-s + 0.288i·12-s − 1.33i·13-s + 0.221·14-s + (−0.258 − 0.516i)15-s + 0.250·16-s + 0.200i·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

\(L(1)\) \(\approx\) \(0.339868 - 1.43970i\)
\(L(\frac12)\) \(\approx\) \(0.339868 - 1.43970i\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.828iT - 7T^{2} \)
11 \( 1 + 0.828T + 11T^{2} \)
13 \( 1 + 4.82iT - 13T^{2} \)
17 \( 1 - 0.828iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 8.48iT - 23T^{2} \)
29 \( 1 + 9.65T + 29T^{2} \)
37 \( 1 + 10.4iT - 37T^{2} \)
41 \( 1 - 7.65T + 41T^{2} \)
43 \( 1 - 9.65iT - 43T^{2} \)
47 \( 1 + 5.65iT - 47T^{2} \)
53 \( 1 + 0.343iT - 53T^{2} \)
59 \( 1 + 3.17T + 59T^{2} \)
61 \( 1 - 0.828T + 61T^{2} \)
67 \( 1 - 9.17iT - 67T^{2} \)
71 \( 1 + 2.82T + 71T^{2} \)
73 \( 1 - 13.6iT - 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 - 1.65iT - 83T^{2} \)
89 \( 1 + 4.82T + 89T^{2} \)
97 \( 1 + 11.3iT - 97T^{2} \)
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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

−9.795041672328777629679201147150, −8.902624786706764002329290319017, −8.260296213432651384168522009522, −7.25873932842218671324122685503, −5.91596310820082940209739079421, −5.52648097532418251507673643851, −4.30221393724165280023682052796, −2.85826137747690753053241121816, −2.07963496605641049375298459077, −0.69654635379179248564905703104, 1.80061650927439254375156084474, 3.29145313993073109638568645818, 4.32924006100929083793666695374, 5.35065236877848828949201858798, 6.06542456016819203145822753348, 7.02100567593980816687052085728, 7.70899178771400918457499132755, 9.095932917174453928544945713277, 9.375070845743693414407770482893, 10.22650087369992168303289824540

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