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Properties

Label 2-2340-39.20-c1-0-15
Degree $2$
Conductor $2340$
Sign $-0.616 + 0.787i$
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 + (1.13 + 0.303i)7-s + (−5.67 + 1.52i)11-s + (1.18 − 3.40i)13-s + (1.54 + 2.67i)17-s + (0.197 − 0.737i)19-s + (0.483 − 0.837i)23-s − 1.00i·25-s + (−1.50 − 0.871i)29-s + (−6.73 − 6.73i)31-s + (−1.01 + 0.586i)35-s + (1.40 + 5.23i)37-s + (1.24 + 4.63i)41-s + (−0.193 + 0.111i)43-s + (−5.91 − 5.91i)47-s + ⋯
L(s)  = 1  + (−0.316 + 0.316i)5-s + (0.428 + 0.114i)7-s + (−1.71 + 0.458i)11-s + (0.329 − 0.944i)13-s + (0.374 + 0.649i)17-s + (0.0453 − 0.169i)19-s + (0.100 − 0.174i)23-s − 0.200i·25-s + (−0.280 − 0.161i)29-s + (−1.20 − 1.20i)31-s + (−0.171 + 0.0991i)35-s + (0.230 + 0.860i)37-s + (0.194 + 0.724i)41-s + (−0.0295 + 0.0170i)43-s + (−0.862 − 0.862i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.616 + 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.616 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.616 + 0.787i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (2321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ -0.616 + 0.787i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5243052474\)
\(L(\frac12)\) \(\approx\) \(0.5243052474\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.18 + 3.40i)T \)
good7 \( 1 + (-1.13 - 0.303i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (5.67 - 1.52i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.54 - 2.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.197 + 0.737i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.483 + 0.837i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.50 + 0.871i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.73 + 6.73i)T + 31iT^{2} \)
37 \( 1 + (-1.40 - 5.23i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.24 - 4.63i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.193 - 0.111i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.91 + 5.91i)T + 47iT^{2} \)
53 \( 1 + 6.79iT - 53T^{2} \)
59 \( 1 + (-2.39 + 8.93i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (5.59 + 9.68i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.68 + 0.451i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.138 - 0.0372i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.76 + 1.76i)T - 73iT^{2} \)
79 \( 1 - 3.45T + 79T^{2} \)
83 \( 1 + (10.0 - 10.0i)T - 83iT^{2} \)
89 \( 1 + (8.76 - 2.34i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.444 + 1.65i)T + (-84.0 - 48.5i)T^{2} \)
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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

−8.327125631793545474140743313249, −8.055987862020021184216151667532, −7.38594533659755415892168998586, −6.35549485426524434593576571302, −5.43096347515167275237597861785, −4.89743820312872667018327496036, −3.71994090049553372705174603911, −2.86390077739220623768791930292, −1.86703603377782716592852198734, −0.17635711767070677994777740669, 1.35232836078001598583929388204, 2.57814448450112091960528877526, 3.54485259746972521104367961899, 4.57644561827618416137968134622, 5.27673871401067781801902528554, 5.98357338812662471174667709434, 7.29291259714133962359228251266, 7.58093578263509549539700212983, 8.554808739626733798794398661580, 9.078443114481461578105783227084

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