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Properties

Label 2-2340-39.5-c1-0-0
Degree $2$
Conductor $2340$
Sign $-0.803 - 0.595i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.803 - 0.595i$
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: \((2,\ 2340,\ (\ :1/2),\ -0.803 - 0.595i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.03359310639\)
\(L(\frac12)\) \(\approx\) \(0.03359310639\)
\(L(\frac{3}{2})\) 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

−9.011736863941229412470427928551, −8.874607320183660986678734764044, −7.60574749447610970950991739339, −7.01542984591982271119736051633, −6.30716437407791566059620591020, −5.35859111180657824151491499748, −4.31896255426267428340946386555, −3.73746033310761156103984338097, −2.69521307682100145152507454682, −1.28775870144655869533123517438, 0.01155997019051056184905293242, 1.88055254319467396438425974827, 2.74245436932077566725405849165, 3.83986397939566879733760739854, 4.54665405889204650271947967818, 5.65561364394717773742593957123, 6.37823960867075090486671787387, 7.16434715872385279285846606898, 7.79129752614623794013551285433, 8.778919043753468668581011279759

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