Properties

Label 2-930-465.464-c1-0-42
Degree $2$
Conductor $930$
Sign $-0.165 + 0.986i$
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 + (−1.02 + 1.39i)3-s + 4-s + (0.467 − 2.18i)5-s + (1.02 − 1.39i)6-s − 1.43i·7-s − 8-s + (−0.884 − 2.86i)9-s + (−0.467 + 2.18i)10-s + 6.14·11-s + (−1.02 + 1.39i)12-s − 2.15·13-s + 1.43i·14-s + (2.56 + 2.90i)15-s + 16-s − 0.937i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.593 + 0.804i)3-s + 0.5·4-s + (0.209 − 0.977i)5-s + (0.419 − 0.568i)6-s − 0.540i·7-s − 0.353·8-s + (−0.294 − 0.955i)9-s + (−0.147 + 0.691i)10-s + 1.85·11-s + (−0.296 + 0.402i)12-s − 0.597·13-s + 0.382i·14-s + (0.662 + 0.748i)15-s + 0.250·16-s − 0.227i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.165 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.165 + 0.986i$
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: \((2,\ 930,\ (\ :1/2),\ -0.165 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.426006 - 0.503267i\)
\(L(\frac12)\) \(\approx\) \(0.426006 - 0.503267i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.02 - 1.39i)T \)
5 \( 1 + (-0.467 + 2.18i)T \)
31 \( 1 + (3.50 + 4.32i)T \)
good7 \( 1 + 1.43iT - 7T^{2} \)
11 \( 1 - 6.14T + 11T^{2} \)
13 \( 1 + 2.15T + 13T^{2} \)
17 \( 1 + 0.937iT - 17T^{2} \)
19 \( 1 + 5.64T + 19T^{2} \)
23 \( 1 + 0.266iT - 23T^{2} \)
29 \( 1 + 3.72T + 29T^{2} \)
37 \( 1 + 7.85T + 37T^{2} \)
41 \( 1 + 3.31iT - 41T^{2} \)
43 \( 1 + 0.581T + 43T^{2} \)
47 \( 1 - 4.74T + 47T^{2} \)
53 \( 1 + 6.07iT - 53T^{2} \)
59 \( 1 + 5.39iT - 59T^{2} \)
61 \( 1 - 9.71iT - 61T^{2} \)
67 \( 1 + 9.90iT - 67T^{2} \)
71 \( 1 - 1.07iT - 71T^{2} \)
73 \( 1 - 3.14T + 73T^{2} \)
79 \( 1 - 2.07iT - 79T^{2} \)
83 \( 1 + 8.35iT - 83T^{2} \)
89 \( 1 - 6.18T + 89T^{2} \)
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

−9.707981560568611656901996864072, −9.119671821052862435541574319492, −8.569267903612820489041018936839, −7.23445252237757582297636828029, −6.41741743646305454709230031263, −5.51425263496275743949736883032, −4.38955871988746177671047730389, −3.76025335151834992380821198455, −1.79070741756194761395723091568, −0.42641063871617895582821232914, 1.55304305165626836674894073710, 2.45016600485651816356438185343, 3.86609689860387349769856900999, 5.44618701421850859439933384138, 6.41530505779953441973877952825, 6.76490676632169323510396627296, 7.61597069740904983940309469279, 8.703396626637600964396989089520, 9.369727170223906776525818950852, 10.48315909132455815682483247729

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