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Properties

Label 2-2340-13.10-c1-0-7
Degree $2$
Conductor $2340$
Sign $-0.111 - 0.993i$
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  + i·5-s + (0.716 + 0.413i)7-s + (−1.5 + 0.866i)11-s + (3.32 + 1.40i)13-s + (−0.716 + 1.24i)17-s + (−0.926 − 0.534i)19-s + (−1.54 − 2.67i)23-s − 25-s + (3.72 + 6.45i)29-s + 5.84i·31-s + (−0.413 + 0.716i)35-s + (0.851 − 0.491i)37-s + (3.69 − 2.13i)41-s + (−4.77 + 8.26i)43-s − 3.46i·47-s + ⋯
L(s)  = 1  + 0.447i·5-s + (0.270 + 0.156i)7-s + (−0.452 + 0.261i)11-s + (0.921 + 0.388i)13-s + (−0.173 + 0.300i)17-s + (−0.212 − 0.122i)19-s + (−0.321 − 0.557i)23-s − 0.200·25-s + (0.692 + 1.19i)29-s + 1.04i·31-s + (−0.0698 + 0.121i)35-s + (0.140 − 0.0808i)37-s + (0.577 − 0.333i)41-s + (−0.727 + 1.26i)43-s − 0.505i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.111 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.111 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.111 - 0.993i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ -0.111 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.492472424\)
\(L(\frac12)\) \(\approx\) \(1.492472424\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
13 \( 1 + (-3.32 - 1.40i)T \)
good7 \( 1 + (-0.716 - 0.413i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.716 - 1.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.926 + 0.534i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.54 + 2.67i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.72 - 6.45i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.84iT - 31T^{2} \)
37 \( 1 + (-0.851 + 0.491i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.69 + 2.13i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.77 - 8.26i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 + 0.334T + 53T^{2} \)
59 \( 1 + (-9.98 - 5.76i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.35 - 2.35i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.9 - 6.87i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.46 + 4.88i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.1iT - 73T^{2} \)
79 \( 1 + 0.252T + 79T^{2} \)
83 \( 1 + 5.67iT - 83T^{2} \)
89 \( 1 + (3.98 - 2.29i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.25 - 4.76i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.935926645869819134144203585884, −8.575483563197329686200170227077, −7.65684226181426620894786227755, −6.80523310672232477459686102012, −6.20896270534881054205435656663, −5.23048305284727206375160097955, −4.39737914334537771727173236673, −3.43317313969706237462568750559, −2.46952003495290443049661934720, −1.36201175617452067309207019492, 0.52591390499410576155984130522, 1.78667913442915785931428109335, 2.95943392459005107521832129308, 3.98583769051537297822773262093, 4.74777787481168326875425096387, 5.73686022966582756774793476018, 6.25706219328745812437322642529, 7.44032657582356457282147483863, 8.066997307563980530087333235036, 8.658494860661687677042523524482

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