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Properties

Label 2-2340-39.5-c1-0-13
Degree $2$
Conductor $2340$
Sign $-0.828 + 0.559i$
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 + (−1.63 − 1.63i)7-s + (−3.24 + 3.24i)11-s + (−3.21 + 1.63i)13-s + 3.72·17-s + (2 − 2i)19-s + 4.17·23-s + 1.00i·25-s − 5.06i·29-s + (−3.26 + 3.26i)31-s − 2.30i·35-s + (−5.21 − 5.21i)37-s + (−7.85 − 7.85i)41-s − 1.35i·43-s + (−4.17 + 4.17i)47-s + ⋯
L(s)  = 1  + (0.316 + 0.316i)5-s + (−0.616 − 0.616i)7-s + (−0.977 + 0.977i)11-s + (−0.891 + 0.452i)13-s + 0.902·17-s + (0.458 − 0.458i)19-s + 0.870·23-s + 0.200i·25-s − 0.941i·29-s + (−0.585 + 0.585i)31-s − 0.389i·35-s + (−0.857 − 0.857i)37-s + (−1.22 − 1.22i)41-s − 0.206i·43-s + (−0.609 + 0.609i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.828 + 0.559i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.828 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.828 + 0.559i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ -0.828 + 0.559i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3388424290\)
\(L(\frac12)\) \(\approx\) \(0.3388424290\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.21 - 1.63i)T \)
good7 \( 1 + (1.63 + 1.63i)T + 7iT^{2} \)
11 \( 1 + (3.24 - 3.24i)T - 11iT^{2} \)
17 \( 1 - 3.72T + 17T^{2} \)
19 \( 1 + (-2 + 2i)T - 19iT^{2} \)
23 \( 1 - 4.17T + 23T^{2} \)
29 \( 1 + 5.06iT - 29T^{2} \)
31 \( 1 + (3.26 - 3.26i)T - 31iT^{2} \)
37 \( 1 + (5.21 + 5.21i)T + 37iT^{2} \)
41 \( 1 + (7.85 + 7.85i)T + 41iT^{2} \)
43 \( 1 + 1.35iT - 43T^{2} \)
47 \( 1 + (4.17 - 4.17i)T - 47iT^{2} \)
53 \( 1 + 11.0iT - 53T^{2} \)
59 \( 1 - 59iT^{2} \)
61 \( 1 + 4.49T + 61T^{2} \)
67 \( 1 + (8.43 - 8.43i)T - 67iT^{2} \)
71 \( 1 + (10.5 + 10.5i)T + 71iT^{2} \)
73 \( 1 + (6.84 + 6.84i)T + 73iT^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + (-5.13 - 5.13i)T + 83iT^{2} \)
89 \( 1 + (-3.61 + 3.61i)T - 89iT^{2} \)
97 \( 1 + (-2.04 + 2.04i)T - 97iT^{2} \)
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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

−8.821156863582773984968706289859, −7.48625485009029936478624804579, −7.34231757972342474622251228508, −6.51335157971467774919813693444, −5.36518579299010899162535575857, −4.84087622597822480300479427066, −3.66618813197110775312340796573, −2.80778637521799192463012301828, −1.80444752645623382066988355731, −0.11074452078518420264372905289, 1.41267750411909984866303240866, 2.94499443510494245177737060581, 3.15312604272038698805903201653, 4.75819236225617172983877181009, 5.48108264245969195590421916026, 5.93480352283380159870433915440, 7.04409548553816861781017512014, 7.83500389156081796400237930547, 8.557403963484007488141618274039, 9.289725392893035193177099481068

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