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Properties

Label 2-2340-260.203-c0-0-1
Degree $2$
Conductor $2340$
Sign $-0.749 + 0.661i$
Analytic cond. $1.16781$
Root an. cond. $1.08065$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − i·5-s − 8-s + i·10-s − 13-s + 16-s + (−1 − i)17-s − i·20-s − 25-s + 26-s − 32-s + (1 + i)34-s − 2i·37-s + i·40-s + (−1 − i)41-s + ⋯
L(s)  = 1  − 2-s + 4-s − i·5-s − 8-s + i·10-s − 13-s + 16-s + (−1 − i)17-s − i·20-s − 25-s + 26-s − 32-s + (1 + i)34-s − 2i·37-s + i·40-s + (−1 − i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.749 + 0.661i$
Analytic conductor: \(1.16781\)
Root analytic conductor: \(1.08065\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :0),\ -0.749 + 0.661i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4307796799\)
\(L(\frac12)\) \(\approx\) \(0.4307796799\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + iT \)
13 \( 1 + T \)
good7 \( 1 + T^{2} \)
11 \( 1 + iT^{2} \)
17 \( 1 + (1 + i)T + iT^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + 2iT - T^{2} \)
41 \( 1 + (1 + i)T + iT^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-1 - i)T + iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (1 + i)T + iT^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−9.032934430102868081961572571755, −8.251158063062505450298638642362, −7.41174570682012903846745105449, −6.88496653765040216983086474049, −5.75349240857321291085900420517, −5.03777940262938604091771763523, −4.03877244749584540619414558117, −2.68858502405261535896790696485, −1.82525860945997193261876228450, −0.37594835211376600312212636205, 1.72661917426311330668462028651, 2.62215576940322222150947620421, 3.47243464191706422808064884128, 4.71740537212222862404869365304, 5.91496627757760657294268990282, 6.68010623081003711136601443782, 7.08796994888002513722604486327, 8.101703104654156672963787340039, 8.534067439751247788244057971772, 9.776131891746952372913377315443

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