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Properties

Label 2-2340-39.32-c1-0-6
Degree $2$
Conductor $2340$
Sign $0.0723 - 0.997i$
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  + (0.707 + 0.707i)5-s + (−0.471 + 1.76i)7-s + (0.870 + 3.24i)11-s + (3.33 + 1.36i)13-s + (−0.219 − 0.380i)17-s + (2.10 + 0.565i)19-s + (2.61 − 4.52i)23-s + 1.00i·25-s + (1.00 + 0.582i)29-s + (1.90 − 1.90i)31-s + (−1.57 + 0.911i)35-s + (−4.97 + 1.33i)37-s + (−10.9 + 2.92i)41-s + (−2.13 + 1.23i)43-s + (0.175 − 0.175i)47-s + ⋯
L(s)  = 1  + (0.316 + 0.316i)5-s + (−0.178 + 0.665i)7-s + (0.262 + 0.979i)11-s + (0.925 + 0.377i)13-s + (−0.0533 − 0.0923i)17-s + (0.483 + 0.129i)19-s + (0.545 − 0.944i)23-s + 0.200i·25-s + (0.187 + 0.108i)29-s + (0.341 − 0.341i)31-s + (−0.266 + 0.154i)35-s + (−0.818 + 0.219i)37-s + (−1.70 + 0.457i)41-s + (−0.326 + 0.188i)43-s + (0.0256 − 0.0256i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0723 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0723 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.0723 - 0.997i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ 0.0723 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.821365920\)
\(L(\frac12)\) \(\approx\) \(1.821365920\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (-3.33 - 1.36i)T \)
good7 \( 1 + (0.471 - 1.76i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.870 - 3.24i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.219 + 0.380i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.10 - 0.565i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.61 + 4.52i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.00 - 0.582i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.90 + 1.90i)T - 31iT^{2} \)
37 \( 1 + (4.97 - 1.33i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (10.9 - 2.92i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.13 - 1.23i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.175 + 0.175i)T - 47iT^{2} \)
53 \( 1 - 10.6iT - 53T^{2} \)
59 \( 1 + (7.48 + 2.00i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-5.54 - 9.60i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.16 + 4.35i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.928 - 3.46i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-7.33 - 7.33i)T + 73iT^{2} \)
79 \( 1 - 5.25T + 79T^{2} \)
83 \( 1 + (8.32 + 8.32i)T + 83iT^{2} \)
89 \( 1 + (1.69 + 6.34i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-18.2 - 4.89i)T + (84.0 + 48.5i)T^{2} \)
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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

−9.065008731796833598270510871825, −8.618233160376897726927538116926, −7.56545013329765207202131796985, −6.72313487762735932921167810847, −6.20483150534836012385904656960, −5.23536679945063008068923119921, −4.41369460939654776338291738365, −3.34770423571465404621664161634, −2.41671459580741797036928710033, −1.38206130664991703042015926891, 0.66391073447278906894014194589, 1.68912484055960553144740190503, 3.26661801353782144808344001867, 3.68087726793889162289209387880, 4.95462474404314493586744020110, 5.63343205271188202942486383114, 6.49726349614953495122072896194, 7.17965689623662591314202814857, 8.239299348350033252761485416531, 8.688732700805479679581055615207

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