Properties

Label 2-1840-5.4-c1-0-61
Degree $2$
Conductor $1840$
Sign $-0.976 - 0.214i$
Analytic cond. $14.6924$
Root an. cond. $3.83307$
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  − 1.95i·3-s + (0.479 − 2.18i)5-s − 2.28i·7-s − 0.840·9-s − 1.12·11-s − 5.95i·13-s + (−4.27 − 0.940i)15-s + 5.80i·17-s − 4.08·19-s − 4.47·21-s i·23-s + (−4.53 − 2.09i)25-s − 4.23i·27-s + 0.408·29-s + 3.19·31-s + ⋯
L(s)  = 1  − 1.13i·3-s + (0.214 − 0.976i)5-s − 0.863i·7-s − 0.280·9-s − 0.338·11-s − 1.65i·13-s + (−1.10 − 0.242i)15-s + 1.40i·17-s − 0.936·19-s − 0.976·21-s − 0.208i·23-s + (−0.907 − 0.419i)25-s − 0.814i·27-s + 0.0758·29-s + 0.573·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.214i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 - 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.976 - 0.214i$
Analytic conductor: \(14.6924\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ -0.976 - 0.214i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.420443796\)
\(L(\frac12)\) \(\approx\) \(1.420443796\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.479 + 2.18i)T \)
23 \( 1 + iT \)
good3 \( 1 + 1.95iT - 3T^{2} \)
7 \( 1 + 2.28iT - 7T^{2} \)
11 \( 1 + 1.12T + 11T^{2} \)
13 \( 1 + 5.95iT - 13T^{2} \)
17 \( 1 - 5.80iT - 17T^{2} \)
19 \( 1 + 4.08T + 19T^{2} \)
29 \( 1 - 0.408T + 29T^{2} \)
31 \( 1 - 3.19T + 31T^{2} \)
37 \( 1 + 9.80iT - 37T^{2} \)
41 \( 1 - 6.27T + 41T^{2} \)
43 \( 1 - 7.75iT - 43T^{2} \)
47 \( 1 - 6.40iT - 47T^{2} \)
53 \( 1 - 6.73iT - 53T^{2} \)
59 \( 1 + 4.75T + 59T^{2} \)
61 \( 1 + 6.33T + 61T^{2} \)
67 \( 1 + 0.283iT - 67T^{2} \)
71 \( 1 - 13.9T + 71T^{2} \)
73 \( 1 + 9.61iT - 73T^{2} \)
79 \( 1 - 4.48T + 79T^{2} \)
83 \( 1 + 10.8iT - 83T^{2} \)
89 \( 1 + 5.68T + 89T^{2} \)
97 \( 1 - 11.0iT - 97T^{2} \)
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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

−8.554989634683303710952969310844, −7.86138124472733723838449056651, −7.57852445648555430477828610495, −6.28964074875278967996998402891, −5.88214651072731595959637212619, −4.70462682363974742598218718957, −3.89054495126938818068238641342, −2.50832698312760097287876785195, −1.39776402200889714222576843490, −0.53046410276135328497254604951, 2.05634196327415386491053317800, 2.86860946908592228367776784242, 3.91709286893344988738954271814, 4.72520866053954903162723754897, 5.52351354884817464765989288068, 6.58154337164540802573051399299, 7.07182369603570402573173533914, 8.302863920604641545441532968283, 9.172684767431059888862369204956, 9.626840171611874520235483444872

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