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Properties

Label 2-2340-65.29-c1-0-2
Degree $2$
Conductor $2340$
Sign $-0.896 - 0.443i$
Analytic cond. $18.6849$
Root an. cond. $4.32261$
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  + (−0.562 + 2.16i)5-s + (−2.51 − 1.44i)7-s + (0.275 + 0.477i)11-s + (1.95 + 3.02i)13-s + (5.90 + 3.40i)17-s + (−1.76 + 3.06i)19-s + (3.45 − 1.99i)23-s + (−4.36 − 2.43i)25-s + (−3.24 − 5.61i)29-s − 6.19·31-s + (4.54 − 4.61i)35-s + (−0.0194 + 0.0112i)37-s + (2.35 + 4.08i)41-s + (−6.26 − 3.61i)43-s − 7.38i·47-s + ⋯
L(s)  = 1  + (−0.251 + 0.967i)5-s + (−0.948 − 0.547i)7-s + (0.0831 + 0.143i)11-s + (0.542 + 0.839i)13-s + (1.43 + 0.826i)17-s + (−0.405 + 0.702i)19-s + (0.719 − 0.415i)23-s + (−0.873 − 0.486i)25-s + (−0.601 − 1.04i)29-s − 1.11·31-s + (0.768 − 0.780i)35-s + (−0.00320 + 0.00184i)37-s + (0.368 + 0.637i)41-s + (−0.955 − 0.551i)43-s − 1.07i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.896 - 0.443i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ -0.896 - 0.443i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7591050823\)
\(L(\frac12)\) \(\approx\) \(0.7591050823\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (0.562 - 2.16i)T \)
13 \( 1 + (-1.95 - 3.02i)T \)
good7 \( 1 + (2.51 + 1.44i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.275 - 0.477i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-5.90 - 3.40i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.76 - 3.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.45 + 1.99i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.24 + 5.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.19T + 31T^{2} \)
37 \( 1 + (0.0194 - 0.0112i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.35 - 4.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.26 + 3.61i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.38iT - 47T^{2} \)
53 \( 1 - 14.1iT - 53T^{2} \)
59 \( 1 + (2.10 - 3.64i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.179 - 0.310i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.3 - 6.55i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.25 - 10.8i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.75iT - 73T^{2} \)
79 \( 1 + 11.9T + 79T^{2} \)
83 \( 1 - 2.96iT - 83T^{2} \)
89 \( 1 + (3.96 + 6.87i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.5 - 7.27i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.425184829959082170959253167900, −8.523745668853793147499306147213, −7.59602504843978259254480577040, −7.03277593948713688278714822635, −6.25363045124457505524974568619, −5.68275962423444280429854039878, −4.11444741004513001150351730784, −3.71938542528748160222225742450, −2.79667226003964624856028402618, −1.49197265963440772821387690155, 0.26601877465254366338085262367, 1.48837325723666803894854612213, 3.06401745256883806350404060639, 3.51101972631384261315219334200, 4.86406598260800539842618888285, 5.45432026037912096735875652652, 6.17079351680096709681507231528, 7.24787833649439908361041835622, 7.897495469586853491419735712288, 8.889775874838060192902946428682

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