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Properties

Label 2-2340-39.11-c1-0-0
Degree $2$
Conductor $2340$
Sign $-0.736 - 0.676i$
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 + (−1.26 − 4.73i)7-s + (−1.02 + 3.82i)11-s + (−1.49 + 3.28i)13-s + (−2.24 + 3.88i)17-s + (1.68 − 0.452i)19-s + (−4.02 − 6.97i)23-s − 1.00i·25-s + (−3.16 + 1.82i)29-s + (0.203 + 0.203i)31-s + (−4.24 − 2.45i)35-s + (−9.64 − 2.58i)37-s + (6.37 + 1.70i)41-s + (−5.53 − 3.19i)43-s + (6.23 + 6.23i)47-s + ⋯
L(s)  = 1  + (0.316 − 0.316i)5-s + (−0.479 − 1.79i)7-s + (−0.308 + 1.15i)11-s + (−0.415 + 0.909i)13-s + (−0.544 + 0.943i)17-s + (0.387 − 0.103i)19-s + (−0.840 − 1.45i)23-s − 0.200i·25-s + (−0.588 + 0.339i)29-s + (0.0365 + 0.0365i)31-s + (−0.718 − 0.414i)35-s + (−1.58 − 0.424i)37-s + (0.996 + 0.266i)41-s + (−0.844 − 0.487i)43-s + (0.909 + 0.909i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.736 - 0.676i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ -0.736 - 0.676i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2004058070\)
\(L(\frac12)\) \(\approx\) \(0.2004058070\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.49 - 3.28i)T \)
good7 \( 1 + (1.26 + 4.73i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.02 - 3.82i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.24 - 3.88i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.68 + 0.452i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (4.02 + 6.97i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.16 - 1.82i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.203 - 0.203i)T + 31iT^{2} \)
37 \( 1 + (9.64 + 2.58i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-6.37 - 1.70i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.53 + 3.19i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.23 - 6.23i)T + 47iT^{2} \)
53 \( 1 + 3.68iT - 53T^{2} \)
59 \( 1 + (-1.42 + 0.381i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.481 - 0.833i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.278 - 1.04i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.67 - 9.96i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (12.0 - 12.0i)T - 73iT^{2} \)
79 \( 1 - 4.50T + 79T^{2} \)
83 \( 1 + (7.49 - 7.49i)T - 83iT^{2} \)
89 \( 1 + (2.91 - 10.8i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.27 + 0.341i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.411834911310397405630878101512, −8.529774525982117072548063288511, −7.57242837752154178115356238967, −6.95837908527393562120440519106, −6.44780565921016061205338972275, −5.20986583899453017477408118315, −4.23412822832897181668759014814, −3.97661481433691025131444345509, −2.43393292580081183514271812166, −1.42427546497265512551083368413, 0.06421007717585502740445245303, 1.96390294191051272253091890692, 2.89034697028248381080073673962, 3.40861763496188476214471674600, 5.03841981221045090936499046603, 5.69701591459254884108202547294, 6.03285933975013095265270134635, 7.18137198497997554256041430318, 8.003171980200048160797641994083, 8.822969143123489907857716231560

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