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Properties

Label 2-2340-12.11-c1-0-17
Degree $2$
Conductor $2340$
Sign $0.225 - 0.974i$
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  + (−1.17 − 0.788i)2-s + (0.756 + 1.85i)4-s + i·5-s + 0.625i·7-s + (0.570 − 2.77i)8-s + (0.788 − 1.17i)10-s + 1.38·11-s − 13-s + (0.492 − 0.733i)14-s + (−2.85 + 2.80i)16-s − 1.76i·17-s + 5.22i·19-s + (−1.85 + 0.756i)20-s + (−1.63 − 1.09i)22-s + 2.64·23-s + ⋯
L(s)  = 1  + (−0.830 − 0.557i)2-s + (0.378 + 0.925i)4-s + 0.447i·5-s + 0.236i·7-s + (0.201 − 0.979i)8-s + (0.249 − 0.371i)10-s + 0.418·11-s − 0.277·13-s + (0.131 − 0.196i)14-s + (−0.713 + 0.700i)16-s − 0.428i·17-s + 1.19i·19-s + (−0.413 + 0.169i)20-s + (−0.347 − 0.233i)22-s + 0.552·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.225 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.225 - 0.974i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1691, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ 0.225 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8916942330\)
\(L(\frac12)\) \(\approx\) \(0.8916942330\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 + 0.788i)T \)
3 \( 1 \)
5 \( 1 - iT \)
13 \( 1 + T \)
good7 \( 1 - 0.625iT - 7T^{2} \)
11 \( 1 - 1.38T + 11T^{2} \)
17 \( 1 + 1.76iT - 17T^{2} \)
19 \( 1 - 5.22iT - 19T^{2} \)
23 \( 1 - 2.64T + 23T^{2} \)
29 \( 1 + 0.138iT - 29T^{2} \)
31 \( 1 - 5.31iT - 31T^{2} \)
37 \( 1 + 1.95T + 37T^{2} \)
41 \( 1 + 2.26iT - 41T^{2} \)
43 \( 1 - 4.50iT - 43T^{2} \)
47 \( 1 - 0.339T + 47T^{2} \)
53 \( 1 + 4.58iT - 53T^{2} \)
59 \( 1 + 1.02T + 59T^{2} \)
61 \( 1 + 8.22T + 61T^{2} \)
67 \( 1 - 0.661iT - 67T^{2} \)
71 \( 1 + 1.23T + 71T^{2} \)
73 \( 1 - 4.49T + 73T^{2} \)
79 \( 1 - 13.2iT - 79T^{2} \)
83 \( 1 - 3.37T + 83T^{2} \)
89 \( 1 - 5.94iT - 89T^{2} \)
97 \( 1 + 3.98T + 97T^{2} \)
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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

−9.236400250289356269254020519179, −8.490973034633113073247102401485, −7.73120913599559141595216588595, −7.00001873472271804509096301957, −6.29904718389951438445533436852, −5.19516663037830268314569314522, −4.02022506275935219675045168609, −3.22323614895241866646082868682, −2.31034128347745556076271572513, −1.21935351492556069632993073564, 0.43915173489741260210682303948, 1.59748511253193290966630894121, 2.75703642450778456185161558605, 4.17286663926048989713579695307, 4.98924616344285586145543547330, 5.83571581877776384582304515710, 6.66636806058638326215626565607, 7.34768356465446118663777166135, 8.059042426994425573697359367427, 8.988550929687078976298758546087

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