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Properties

Label 2-2340-65.57-c1-0-13
Degree $2$
Conductor $2340$
Sign $0.966 - 0.256i$
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  − 2.23i·5-s + (2.61 + 2.61i)11-s + (−2 + 3i)13-s + (2.23 − 2.23i)17-s + (5.85 + 5.85i)19-s + (−3.38 − 3.38i)23-s − 5.00·25-s + 5.23i·29-s + (5.85 − 5.85i)31-s + 9.70i·37-s + (−3.76 + 3.76i)41-s + (−3.85 − 3.85i)43-s + 8.94i·47-s + 7·49-s + (−1.47 + 1.47i)53-s + ⋯
L(s)  = 1  − 0.999i·5-s + (0.789 + 0.789i)11-s + (−0.554 + 0.832i)13-s + (0.542 − 0.542i)17-s + (1.34 + 1.34i)19-s + (−0.705 − 0.705i)23-s − 1.00·25-s + 0.972i·29-s + (1.05 − 1.05i)31-s + 1.59i·37-s + (−0.587 + 0.587i)41-s + (−0.587 − 0.587i)43-s + 1.30i·47-s + 49-s + (−0.202 + 0.202i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.966 - 0.256i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ 0.966 - 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.821891242\)
\(L(\frac12)\) \(\approx\) \(1.821891242\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23iT \)
13 \( 1 + (2 - 3i)T \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + (-2.61 - 2.61i)T + 11iT^{2} \)
17 \( 1 + (-2.23 + 2.23i)T - 17iT^{2} \)
19 \( 1 + (-5.85 - 5.85i)T + 19iT^{2} \)
23 \( 1 + (3.38 + 3.38i)T + 23iT^{2} \)
29 \( 1 - 5.23iT - 29T^{2} \)
31 \( 1 + (-5.85 + 5.85i)T - 31iT^{2} \)
37 \( 1 - 9.70iT - 37T^{2} \)
41 \( 1 + (3.76 - 3.76i)T - 41iT^{2} \)
43 \( 1 + (3.85 + 3.85i)T + 43iT^{2} \)
47 \( 1 - 8.94iT - 47T^{2} \)
53 \( 1 + (1.47 - 1.47i)T - 53iT^{2} \)
59 \( 1 + (-3.38 + 3.38i)T - 59iT^{2} \)
61 \( 1 - 5.70T + 61T^{2} \)
67 \( 1 + 0.291T + 67T^{2} \)
71 \( 1 + (2.61 - 2.61i)T - 71iT^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 + 3.70iT - 79T^{2} \)
83 \( 1 + 10.4iT - 83T^{2} \)
89 \( 1 + (-2.23 + 2.23i)T - 89iT^{2} \)
97 \( 1 - 15.4T + 97T^{2} \)
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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

−9.101498473155164723825307185131, −8.234005103817220782302507103913, −7.58535115007675060857447000572, −6.71627340147355089720103066582, −5.85168947634319406147281034353, −4.90308086578868861123383054332, −4.36850045841657432318850011617, −3.36109205748141406254339579891, −1.98221446923143905892971355481, −1.08255356534435903536922387422, 0.75663905842047902556270462590, 2.28236865222397606820576494257, 3.24052041124100521842339830390, 3.81923703491424639438027110711, 5.17648650337015644912133188417, 5.83517397011830226209491309634, 6.70432109003418189178910977831, 7.37633014870060540059042164465, 8.086949198950060279147068409776, 8.982313368259731453917825392935

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