↑

Properties

Label 2-2340-39.5-c1-0-4
Degree $2$
Conductor $2340$
Sign $-0.564 - 0.825i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)5-s + (2.50 + 2.50i)7-s + (−1.31 + 1.31i)11-s + (1.64 + 3.20i)13-s − 3.11·17-s − 3.74·23-s + 1.00i·25-s + 4.03i·29-s + (−3.29 + 3.29i)31-s − 3.54i·35-s + (−2.94 − 2.94i)37-s + (0.525 + 0.525i)41-s + 0.288i·43-s + (−1.90 + 1.90i)47-s + 5.55i·49-s + ⋯
L(s)  = 1  + (−0.316 − 0.316i)5-s + (0.947 + 0.947i)7-s + (−0.395 + 0.395i)11-s + (0.457 + 0.889i)13-s − 0.756·17-s − 0.781·23-s + 0.200i·25-s + 0.749i·29-s + (−0.592 + 0.592i)31-s − 0.598i·35-s + (−0.484 − 0.484i)37-s + (0.0821 + 0.0821i)41-s + 0.0439i·43-s + (−0.278 + 0.278i)47-s + 0.793i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.564 - 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.564 - 0.825i$
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: \((2,\ 2340,\ (\ :1/2),\ -0.564 - 0.825i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.116515330\)
\(L(\frac12)\) \(\approx\) \(1.116515330\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 - 3.20i)T \)
good7 \( 1 + (-2.50 - 2.50i)T + 7iT^{2} \)
11 \( 1 + (1.31 - 1.31i)T - 11iT^{2} \)
17 \( 1 + 3.11T + 17T^{2} \)
19 \( 1 - 19iT^{2} \)
23 \( 1 + 3.74T + 23T^{2} \)
29 \( 1 - 4.03iT - 29T^{2} \)
31 \( 1 + (3.29 - 3.29i)T - 31iT^{2} \)
37 \( 1 + (2.94 + 2.94i)T + 37iT^{2} \)
41 \( 1 + (-0.525 - 0.525i)T + 41iT^{2} \)
43 \( 1 - 0.288iT - 43T^{2} \)
47 \( 1 + (1.90 - 1.90i)T - 47iT^{2} \)
53 \( 1 + 11.4iT - 53T^{2} \)
59 \( 1 + (5.24 - 5.24i)T - 59iT^{2} \)
61 \( 1 + 4.04T + 61T^{2} \)
67 \( 1 + (3.01 - 3.01i)T - 67iT^{2} \)
71 \( 1 + (-1.10 - 1.10i)T + 71iT^{2} \)
73 \( 1 + (-5.26 - 5.26i)T + 73iT^{2} \)
79 \( 1 + 5.75T + 79T^{2} \)
83 \( 1 + (2.55 + 2.55i)T + 83iT^{2} \)
89 \( 1 + (1.09 - 1.09i)T - 89iT^{2} \)
97 \( 1 + (-4.66 + 4.66i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.911786811888655183077201874534, −8.685416704734005499337823544054, −7.81215729520387626622559179829, −7.00735525155338297667252148080, −6.08310780205112341027251095543, −5.18056316476637517658435756005, −4.59883560090267966464318922454, −3.64362389015615023235379468523, −2.31355981957411568101802253437, −1.58366534634477908255822078751, 0.36800061843983229353483439457, 1.70387467559961413143476418293, 2.93211451414692583326426286721, 3.91908596018754210692512192588, 4.59276830828834099353051802661, 5.59144458919881181208498939574, 6.39630297180669397022223645974, 7.40407112975803254308990855649, 7.916086545613978664392440928226, 8.477294564258073985618368357675

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