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Properties

Label 2-1840-460.459-c1-0-61
Degree $2$
Conductor $1840$
Sign $-0.0522 - 0.998i$
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  − 2.49·3-s + (−2.19 + 0.417i)5-s − 3.04i·7-s + 3.24·9-s − 5.87·11-s − 6.00i·13-s + (5.48 − 1.04i)15-s − 0.968·17-s − 1.72·19-s + 7.61i·21-s + (−3.79 − 2.93i)23-s + (4.65 − 1.83i)25-s − 0.611·27-s − 3.89·29-s − 9.01i·31-s + ⋯
L(s)  = 1  − 1.44·3-s + (−0.982 + 0.186i)5-s − 1.15i·7-s + 1.08·9-s − 1.77·11-s − 1.66i·13-s + (1.41 − 0.269i)15-s − 0.234·17-s − 0.395·19-s + 1.66i·21-s + (−0.790 − 0.612i)23-s + (0.930 − 0.366i)25-s − 0.117·27-s − 0.723·29-s − 1.61i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0522 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0522 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.0522 - 0.998i$
Analytic conductor: \(14.6924\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1839, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ -0.0522 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.07773047811\)
\(L(\frac12)\) \(\approx\) \(0.07773047811\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 - 0.417i)T \)
23 \( 1 + (3.79 + 2.93i)T \)
good3 \( 1 + 2.49T + 3T^{2} \)
7 \( 1 + 3.04iT - 7T^{2} \)
11 \( 1 + 5.87T + 11T^{2} \)
13 \( 1 + 6.00iT - 13T^{2} \)
17 \( 1 + 0.968T + 17T^{2} \)
19 \( 1 + 1.72T + 19T^{2} \)
29 \( 1 + 3.89T + 29T^{2} \)
31 \( 1 + 9.01iT - 31T^{2} \)
37 \( 1 - 2.15T + 37T^{2} \)
41 \( 1 - 6.17T + 41T^{2} \)
43 \( 1 + 9.08iT - 43T^{2} \)
47 \( 1 + 9.73T + 47T^{2} \)
53 \( 1 + 9.18T + 53T^{2} \)
59 \( 1 - 1.17iT - 59T^{2} \)
61 \( 1 + 0.951iT - 61T^{2} \)
67 \( 1 - 9.83iT - 67T^{2} \)
71 \( 1 + 8.99iT - 71T^{2} \)
73 \( 1 - 8.53iT - 73T^{2} \)
79 \( 1 + 5.59T + 79T^{2} \)
83 \( 1 - 5.97iT - 83T^{2} \)
89 \( 1 - 7.22iT - 89T^{2} \)
97 \( 1 + 7.51T + 97T^{2} \)
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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

−8.243584935765568905749363242575, −7.79359655989232759672509722197, −7.18289591901112218422091624184, −6.14762643812793846917425826101, −5.42736209649192702100577739926, −4.64106357371084057966765396862, −3.82035045033909276892919543271, −2.64129419842986683363158689457, −0.58209562456674250556856457191, −0.06532892183612302590474994814, 1.80644240610183202591737178963, 3.07827045400599393820689247418, 4.48343477897121511692159230438, 4.94722805858277034998300920446, 5.79366087503849422610007739726, 6.49910635310289200803146436962, 7.42178185301083086271332702218, 8.212628572437510891562718301169, 9.014570903261871086222783752830, 9.910319189029532329768416445764

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