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Properties

Label 2-2340-39.8-c1-0-0
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: $\chi_{2340} (1061, \cdot )$
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

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

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