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Properties

Label 2-2394-133.132-c1-0-53
Degree $2$
Conductor $2394$
Sign $-0.433 + 0.900i$
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 − 0.485i·5-s + (2.59 − 0.523i)7-s + i·8-s − 0.485·10-s + 0.891·11-s + 2.48·13-s + (−0.523 − 2.59i)14-s + 16-s − 5.60i·17-s + (−3.47 − 2.63i)19-s + 0.485i·20-s − 0.891i·22-s − 7.41·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 0.217i·5-s + (0.980 − 0.197i)7-s + 0.353i·8-s − 0.153·10-s + 0.268·11-s + 0.689·13-s + (−0.139 − 0.693i)14-s + 0.250·16-s − 1.36i·17-s + (−0.797 − 0.603i)19-s + 0.108i·20-s − 0.189i·22-s − 1.54·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.433 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.433 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2394\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $-0.433 + 0.900i$
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: \((2,\ 2394,\ (\ :1/2),\ -0.433 + 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.807782875\)
\(L(\frac12)\) \(\approx\) \(1.807782875\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 + 0.523i)T \)
19 \( 1 + (3.47 + 2.63i)T \)
good5 \( 1 + 0.485iT - 5T^{2} \)
11 \( 1 - 0.891T + 11T^{2} \)
13 \( 1 - 2.48T + 13T^{2} \)
17 \( 1 + 5.60iT - 17T^{2} \)
23 \( 1 + 7.41T + 23T^{2} \)
29 \( 1 - 2.73iT - 29T^{2} \)
31 \( 1 - 2.14T + 31T^{2} \)
37 \( 1 - 5.04iT - 37T^{2} \)
41 \( 1 - 7.37T + 41T^{2} \)
43 \( 1 - 0.621T + 43T^{2} \)
47 \( 1 + 1.33iT - 47T^{2} \)
53 \( 1 + 12.4iT - 53T^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 + 5.88iT - 61T^{2} \)
67 \( 1 - 8.29iT - 67T^{2} \)
71 \( 1 - 5.99iT - 71T^{2} \)
73 \( 1 + 9.16iT - 73T^{2} \)
79 \( 1 + 12.9iT - 79T^{2} \)
83 \( 1 + 17.4iT - 83T^{2} \)
89 \( 1 + 9.70T + 89T^{2} \)
97 \( 1 + 2.53T + 97T^{2} \)
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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

−8.682904475146224795249865018297, −8.233310535969129532895686896501, −7.26782434385824583938340398017, −6.38335967758728108461087650704, −5.30158403370930212025149109393, −4.62734383123718887113536796331, −3.90053484067691269553946149982, −2.75527923046149414320372984202, −1.77416416021199379094903334383, −0.67825801151998260735192381810, 1.29317615585829722388053682998, 2.39079373152533740045139731276, 3.98892850373370511585104545199, 4.23415287919611165219115125047, 5.58956157175589076376842204704, 6.02197137916827285353340362566, 6.83418016533711550008780735700, 7.901851086616638591107619608284, 8.258292800704208923554639691328, 8.916049008765643816933393257527

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