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Properties

Label 2-2340-15.8-c1-0-1
Degree $2$
Conductor $2340$
Sign $-0.977 + 0.213i$
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  + (1.25 + 1.84i)5-s + (−1.73 + 1.73i)7-s + 0.384i·11-s + (−0.707 − 0.707i)13-s + (−4.01 − 4.01i)17-s + 1.31i·19-s + (−3.09 + 3.09i)23-s + (−1.82 + 4.65i)25-s − 5.58·29-s − 4.45·31-s + (−5.38 − 1.01i)35-s + (7.95 − 7.95i)37-s + 7.37i·41-s + (−6.63 − 6.63i)43-s + (9.03 + 9.03i)47-s + ⋯
L(s)  = 1  + (0.563 + 0.826i)5-s + (−0.655 + 0.655i)7-s + 0.116i·11-s + (−0.196 − 0.196i)13-s + (−0.972 − 0.972i)17-s + 0.302i·19-s + (−0.644 + 0.644i)23-s + (−0.365 + 0.930i)25-s − 1.03·29-s − 0.800·31-s + (−0.910 − 0.172i)35-s + (1.30 − 1.30i)37-s + 1.15i·41-s + (−1.01 − 1.01i)43-s + (1.31 + 1.31i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.977 + 0.213i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ -0.977 + 0.213i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3801997460\)
\(L(\frac12)\) \(\approx\) \(0.3801997460\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.25 - 1.84i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (1.73 - 1.73i)T - 7iT^{2} \)
11 \( 1 - 0.384iT - 11T^{2} \)
17 \( 1 + (4.01 + 4.01i)T + 17iT^{2} \)
19 \( 1 - 1.31iT - 19T^{2} \)
23 \( 1 + (3.09 - 3.09i)T - 23iT^{2} \)
29 \( 1 + 5.58T + 29T^{2} \)
31 \( 1 + 4.45T + 31T^{2} \)
37 \( 1 + (-7.95 + 7.95i)T - 37iT^{2} \)
41 \( 1 - 7.37iT - 41T^{2} \)
43 \( 1 + (6.63 + 6.63i)T + 43iT^{2} \)
47 \( 1 + (-9.03 - 9.03i)T + 47iT^{2} \)
53 \( 1 + (-4.78 + 4.78i)T - 53iT^{2} \)
59 \( 1 + 8.10T + 59T^{2} \)
61 \( 1 + 15.0T + 61T^{2} \)
67 \( 1 + (-5.47 + 5.47i)T - 67iT^{2} \)
71 \( 1 - 1.23iT - 71T^{2} \)
73 \( 1 + (7.10 + 7.10i)T + 73iT^{2} \)
79 \( 1 + 11.1iT - 79T^{2} \)
83 \( 1 + (5.98 - 5.98i)T - 83iT^{2} \)
89 \( 1 + 10.1T + 89T^{2} \)
97 \( 1 + (-3.76 + 3.76i)T - 97iT^{2} \)
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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

−9.435097783820251443556186608005, −8.902067191058409891573723287939, −7.59360394143198211743488788863, −7.18934076308790048672154033759, −6.05522319642837731529435406915, −5.83610010666624372543342211516, −4.65776303413840065397328924726, −3.52137408127553158593119161817, −2.69170386299422299806290375714, −1.89759113350179022518698999342, 0.11905110093829531305702870738, 1.51173254530528671989079972434, 2.54655456754788449812176368726, 3.88133679805467744566974219511, 4.43065325744909141474826308654, 5.49279005220929381920442902343, 6.24295323835338804659817590239, 6.92070521376264289820216634900, 7.88832929915591097610531611202, 8.718164557204105246047189206611

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