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Properties

Label 2-2340-39.5-c1-0-6
Degree $2$
Conductor $2340$
Sign $0.988 - 0.150i$
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 + (−2.78 − 2.78i)7-s + (−0.479 + 0.479i)11-s + (−1.10 + 3.43i)13-s + 3.43·17-s − 0.154·23-s + 1.00i·25-s − 0.454i·29-s + (2.21 − 2.21i)31-s − 3.94i·35-s + (5.32 + 5.32i)37-s + (5.48 + 5.48i)41-s + 5.35i·43-s + (5.81 − 5.81i)47-s + 8.53i·49-s + ⋯
L(s)  = 1  + (0.316 + 0.316i)5-s + (−1.05 − 1.05i)7-s + (−0.144 + 0.144i)11-s + (−0.307 + 0.951i)13-s + 0.833·17-s − 0.0321·23-s + 0.200i·25-s − 0.0844i·29-s + (0.398 − 0.398i)31-s − 0.666i·35-s + (0.875 + 0.875i)37-s + (0.856 + 0.856i)41-s + 0.816i·43-s + (0.847 − 0.847i)47-s + 1.21i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.150i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.988 - 0.150i$
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: \((2,\ 2340,\ (\ :1/2),\ 0.988 - 0.150i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.572999370\)
\(L(\frac12)\) \(\approx\) \(1.572999370\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.10 - 3.43i)T \)
good7 \( 1 + (2.78 + 2.78i)T + 7iT^{2} \)
11 \( 1 + (0.479 - 0.479i)T - 11iT^{2} \)
17 \( 1 - 3.43T + 17T^{2} \)
19 \( 1 - 19iT^{2} \)
23 \( 1 + 0.154T + 23T^{2} \)
29 \( 1 + 0.454iT - 29T^{2} \)
31 \( 1 + (-2.21 + 2.21i)T - 31iT^{2} \)
37 \( 1 + (-5.32 - 5.32i)T + 37iT^{2} \)
41 \( 1 + (-5.48 - 5.48i)T + 41iT^{2} \)
43 \( 1 - 5.35iT - 43T^{2} \)
47 \( 1 + (-5.81 + 5.81i)T - 47iT^{2} \)
53 \( 1 + 11.6iT - 53T^{2} \)
59 \( 1 + (1.91 - 1.91i)T - 59iT^{2} \)
61 \( 1 - 9.97T + 61T^{2} \)
67 \( 1 + (-7.57 + 7.57i)T - 67iT^{2} \)
71 \( 1 + (-4.26 - 4.26i)T + 71iT^{2} \)
73 \( 1 + (-3.18 - 3.18i)T + 73iT^{2} \)
79 \( 1 - 13.3T + 79T^{2} \)
83 \( 1 + (12.7 + 12.7i)T + 83iT^{2} \)
89 \( 1 + (-10.6 + 10.6i)T - 89iT^{2} \)
97 \( 1 + (8.68 - 8.68i)T - 97iT^{2} \)
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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

−9.251330735341178300179288326275, −8.107065932206462852068086767388, −7.37364872359636694160488954544, −6.65125380565830535835470566477, −6.13633842870339595847108912738, −4.98419631960606131687585006081, −4.06252777439459077471468609486, −3.29451558190515219709496188505, −2.26635536907510033669641495810, −0.852732926387324783470819514247, 0.75333384362968107670387485333, 2.36432674546193336020127266181, 3.00738727523481614221251135017, 4.04372903993043775993339524583, 5.39801612568752141687753001331, 5.64434819563664076110487236536, 6.50681941099254567253495882064, 7.50388744804066955496696363251, 8.257616638223795021186744122595, 9.120762979568782000080822983691

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