Properties

Label 2-4140-5.4-c1-0-6
Degree $2$
Conductor $4140$
Sign $0.606 - 0.794i$
Analytic cond. $33.0580$
Root an. cond. $5.74961$
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  + (−1.77 − 1.35i)5-s − 3.32i·7-s − 5.77·11-s + 1.10i·13-s + 0.893i·17-s − 2.42·19-s i·23-s + (1.31 + 4.82i)25-s + 4.11·29-s − 9.54·31-s + (−4.50 + 5.90i)35-s + 7.69i·37-s − 0.00418·41-s − 9.97i·43-s + 10.0i·47-s + ⋯
L(s)  = 1  + (−0.794 − 0.606i)5-s − 1.25i·7-s − 1.74·11-s + 0.305i·13-s + 0.216i·17-s − 0.557·19-s − 0.208i·23-s + (0.263 + 0.964i)25-s + 0.763·29-s − 1.71·31-s + (−0.761 + 0.998i)35-s + 1.26i·37-s − 0.000653·41-s − 1.52i·43-s + 1.45i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.606 - 0.794i$
Analytic conductor: \(33.0580\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ 0.606 - 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5563544836\)
\(L(\frac12)\) \(\approx\) \(0.5563544836\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.77 + 1.35i)T \)
23 \( 1 + iT \)
good7 \( 1 + 3.32iT - 7T^{2} \)
11 \( 1 + 5.77T + 11T^{2} \)
13 \( 1 - 1.10iT - 13T^{2} \)
17 \( 1 - 0.893iT - 17T^{2} \)
19 \( 1 + 2.42T + 19T^{2} \)
29 \( 1 - 4.11T + 29T^{2} \)
31 \( 1 + 9.54T + 31T^{2} \)
37 \( 1 - 7.69iT - 37T^{2} \)
41 \( 1 + 0.00418T + 41T^{2} \)
43 \( 1 + 9.97iT - 43T^{2} \)
47 \( 1 - 10.0iT - 47T^{2} \)
53 \( 1 + 6.25iT - 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 - 10.5T + 61T^{2} \)
67 \( 1 + 10.9iT - 67T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 - 1.89iT - 73T^{2} \)
79 \( 1 - 0.216T + 79T^{2} \)
83 \( 1 - 5.38iT - 83T^{2} \)
89 \( 1 - 6.00T + 89T^{2} \)
97 \( 1 + 2.08iT - 97T^{2} \)
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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

−8.290052661381125365370832181909, −7.84835756010047419934040502931, −7.24244651472669985789489709621, −6.47270111037637694013060693304, −5.30265341679439430954647580532, −4.77789877084746810222212371878, −3.98694464672688339171885106700, −3.26561260070428359127523160374, −2.05936558929667199786509309101, −0.76587857142452346409197521986, 0.21568505222665742889237894147, 2.15725487853367412254855041578, 2.73994484749419827041977251176, 3.53330745001436523902212700592, 4.62815233714179590657830623968, 5.42051974093039052573193270882, 5.95453683818825481135351221303, 6.99985755132390564941598332662, 7.64117168361458312718228451491, 8.263401251778805061510319385799

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