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Properties

Label 2-7644-13.12-c1-0-71
Degree $2$
Conductor $7644$
Sign $0.973 - 0.228i$
Analytic cond. $61.0376$
Root an. cond. $7.81265$
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  + 3-s + 3.05i·5-s + 9-s − 0.989i·11-s + (3.51 − 0.822i)13-s + 3.05i·15-s + 7.83·17-s − 4.04i·19-s + 2.51·23-s − 4.32·25-s + 27-s − 5.32·29-s + 5.68i·31-s − 0.989i·33-s − 6.44i·37-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.36i·5-s + 0.333·9-s − 0.298i·11-s + (0.973 − 0.228i)13-s + 0.788i·15-s + 1.90·17-s − 0.927i·19-s + 0.523·23-s − 0.864·25-s + 0.192·27-s − 0.988·29-s + 1.02i·31-s − 0.172i·33-s − 1.05i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.228i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7644\)    =    \(2^{2} \cdot 3 \cdot 7^{2} \cdot 13\)
Sign: $0.973 - 0.228i$
Analytic conductor: \(61.0376\)
Root analytic conductor: \(7.81265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7644} (4705, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7644,\ (\ :1/2),\ 0.973 - 0.228i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.052570801\)
\(L(\frac12)\) \(\approx\) \(3.052570801\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
13 \( 1 + (-3.51 + 0.822i)T \)
good5 \( 1 - 3.05iT - 5T^{2} \)
11 \( 1 + 0.989iT - 11T^{2} \)
17 \( 1 - 7.83T + 17T^{2} \)
19 \( 1 + 4.04iT - 19T^{2} \)
23 \( 1 - 2.51T + 23T^{2} \)
29 \( 1 + 5.32T + 29T^{2} \)
31 \( 1 - 5.68iT - 31T^{2} \)
37 \( 1 + 6.44iT - 37T^{2} \)
41 \( 1 + 9.07iT - 41T^{2} \)
43 \( 1 - 3.32T + 43T^{2} \)
47 \( 1 + 12.7iT - 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 - 8.74iT - 59T^{2} \)
61 \( 1 - 8.64T + 61T^{2} \)
67 \( 1 - 9.73iT - 67T^{2} \)
71 \( 1 + 16.4iT - 71T^{2} \)
73 \( 1 + 10.1iT - 73T^{2} \)
79 \( 1 + 0.302T + 79T^{2} \)
83 \( 1 + 4.69iT - 83T^{2} \)
89 \( 1 + 0.235iT - 89T^{2} \)
97 \( 1 + 12.4iT - 97T^{2} \)
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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

−7.70368975504142311646392190764, −7.24715520591371102850795302410, −6.72928538053147438004340611470, −5.74430156597315885799077399175, −5.33734354066275183893118826654, −3.94611578687377420906179096214, −3.45252453172703660285023463831, −2.90042123133004814273304181947, −2.00201374380402857954869237178, −0.809767138864285561880919689096, 1.07806923267706044368939791617, 1.43081600644455772771100020773, 2.68242525899218498931692589669, 3.67967924718778127577393161869, 4.13446395909953153709004466694, 5.09054566368277140147614259604, 5.63158602951124158140421815060, 6.37062703063982651557182175809, 7.41313458408957653982490030752, 8.108927341338640645074963324293

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