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Properties

Label 2-920-5.4-c1-0-30
Degree $2$
Conductor $920$
Sign $-0.673 - 0.739i$
Analytic cond. $7.34623$
Root an. cond. $2.71039$
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  − 2.89i·3-s + (−1.50 − 1.65i)5-s + 0.580i·7-s − 5.40·9-s − 0.0809·11-s − 3.02i·13-s + (−4.79 + 4.36i)15-s − 0.280i·17-s − 4.72·19-s + 1.68·21-s + i·23-s + (−0.470 + 4.97i)25-s + 6.98i·27-s − 1.38·29-s − 5.70·31-s + ⋯
L(s)  = 1  − 1.67i·3-s + (−0.673 − 0.739i)5-s + 0.219i·7-s − 1.80·9-s − 0.0243·11-s − 0.837i·13-s + (−1.23 + 1.12i)15-s − 0.0680i·17-s − 1.08·19-s + 0.367·21-s + 0.208i·23-s + (−0.0940 + 0.995i)25-s + 1.34i·27-s − 0.256·29-s − 1.02·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.673 - 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.673 - 0.739i$
Analytic conductor: \(7.34623\)
Root analytic conductor: \(2.71039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :1/2),\ -0.673 - 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.263006 + 0.594912i\)
\(L(\frac12)\) \(\approx\) \(0.263006 + 0.594912i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.50 + 1.65i)T \)
23 \( 1 - iT \)
good3 \( 1 + 2.89iT - 3T^{2} \)
7 \( 1 - 0.580iT - 7T^{2} \)
11 \( 1 + 0.0809T + 11T^{2} \)
13 \( 1 + 3.02iT - 13T^{2} \)
17 \( 1 + 0.280iT - 17T^{2} \)
19 \( 1 + 4.72T + 19T^{2} \)
29 \( 1 + 1.38T + 29T^{2} \)
31 \( 1 + 5.70T + 31T^{2} \)
37 \( 1 + 2.61iT - 37T^{2} \)
41 \( 1 - 5.31T + 41T^{2} \)
43 \( 1 + 7.30iT - 43T^{2} \)
47 \( 1 - 12.2iT - 47T^{2} \)
53 \( 1 - 6.64iT - 53T^{2} \)
59 \( 1 + 3.70T + 59T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 + 5.99iT - 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + 1.14iT - 73T^{2} \)
79 \( 1 - 1.68T + 79T^{2} \)
83 \( 1 + 9.86iT - 83T^{2} \)
89 \( 1 + 6.69T + 89T^{2} \)
97 \( 1 + 12.8iT - 97T^{2} \)
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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

−9.167213684582974392872057827820, −8.539257660097620878937470992715, −7.71315631758972242787420940457, −7.27274976111664464097785558415, −6.12323799682489271463934629646, −5.41914637849643281994933509527, −4.09608310547633295914565484978, −2.74498339185189820922124139413, −1.55841226145620979598485741212, −0.30376559036992195216469201814, 2.48810551028589288751792343147, 3.74329851962477145076990664930, 4.12784064604449215875548657385, 5.12633346697529769387180988266, 6.28130811978136145731494996247, 7.20009999622583674813040586676, 8.340789337133373128053362652484, 9.039492083810202413842914231624, 9.950928915229073112702744331861, 10.54834030967595575749563423885

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