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Properties

Label 2-2394-133.132-c1-0-44
Degree $2$
Conductor $2394$
Sign $0.790 + 0.612i$
Analytic cond. $19.1161$
Root an. cond. $4.37220$
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  + i·2-s − 4-s − 2.68i·5-s + (−0.592 − 2.57i)7-s − i·8-s + 2.68·10-s + 1.22·11-s + 5.01·13-s + (2.57 − 0.592i)14-s + 16-s + 5.15i·17-s + (2.75 + 3.37i)19-s + 2.68i·20-s + 1.22i·22-s − 3.18·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 1.20i·5-s + (−0.224 − 0.974i)7-s − 0.353i·8-s + 0.850·10-s + 0.370·11-s + 1.39·13-s + (0.689 − 0.158i)14-s + 0.250·16-s + 1.25i·17-s + (0.632 + 0.774i)19-s + 0.601i·20-s + 0.261i·22-s − 0.664·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2394\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $0.790 + 0.612i$
Analytic conductor: \(19.1161\)
Root analytic conductor: \(4.37220\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2394} (1063, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2394,\ (\ :1/2),\ 0.790 + 0.612i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.740022438\)
\(L(\frac12)\) \(\approx\) \(1.740022438\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 + (0.592 + 2.57i)T \)
19 \( 1 + (-2.75 - 3.37i)T \)
good5 \( 1 + 2.68iT - 5T^{2} \)
11 \( 1 - 1.22T + 11T^{2} \)
13 \( 1 - 5.01T + 13T^{2} \)
17 \( 1 - 5.15iT - 17T^{2} \)
23 \( 1 + 3.18T + 23T^{2} \)
29 \( 1 + 4.47iT - 29T^{2} \)
31 \( 1 - 6.61T + 31T^{2} \)
37 \( 1 + 3.74iT - 37T^{2} \)
41 \( 1 + 0.873T + 41T^{2} \)
43 \( 1 - 8.11T + 43T^{2} \)
47 \( 1 + 7.36iT - 47T^{2} \)
53 \( 1 + 1.58iT - 53T^{2} \)
59 \( 1 + 2.46T + 59T^{2} \)
61 \( 1 + 9.69iT - 61T^{2} \)
67 \( 1 + 0.728iT - 67T^{2} \)
71 \( 1 + 5.14iT - 71T^{2} \)
73 \( 1 + 6.11iT - 73T^{2} \)
79 \( 1 + 9.74iT - 79T^{2} \)
83 \( 1 - 1.81iT - 83T^{2} \)
89 \( 1 - 3.38T + 89T^{2} \)
97 \( 1 + 3.22T + 97T^{2} \)
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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

−8.664329472606072731055174102483, −8.155122205253615465580646353919, −7.51357466821607239206638263477, −6.29168034783868138963795395016, −6.03006137966304597486956127251, −4.91986563863589414301274579033, −4.04153858530846477850488372961, −3.63163179265681936806874709746, −1.60738367840381044228248959346, −0.71618471907976159812304710205, 1.17964508129986407415835960311, 2.64554228070615031583674013615, 2.95421513321509348678033801603, 3.96881240903079475051872218963, 5.06613142454894455390251487726, 6.00604491081909985877838425206, 6.62552737105462077008965913986, 7.49315454255371385024660604070, 8.525354081695072097981355705539, 9.132696036943238403228816549665

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