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Properties

Label 2-1840-460.459-c1-0-21
Degree $2$
Conductor $1840$
Sign $0.295 - 0.955i$
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  + 2.23·3-s + (−1.58 − 1.58i)5-s + 2.82i·7-s + 2.00·9-s + 5i·13-s + (−3.53 − 3.53i)15-s + 3.16·17-s − 7.07·19-s + 6.32i·21-s + (2.23 + 4.24i)23-s + 5.00i·25-s − 2.23·27-s + 9·29-s + 6.70i·31-s + (4.47 − 4.47i)35-s + ⋯
L(s)  = 1  + 1.29·3-s + (−0.707 − 0.707i)5-s + 1.06i·7-s + 0.666·9-s + 1.38i·13-s + (−0.912 − 0.912i)15-s + 0.766·17-s − 1.62·19-s + 1.38i·21-s + (0.466 + 0.884i)23-s + 1.00i·25-s − 0.430·27-s + 1.67·29-s + 1.20i·31-s + (0.755 − 0.755i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.295 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.295 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.295 - 0.955i$
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: \((2,\ 1840,\ (\ :1/2),\ 0.295 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.032659278\)
\(L(\frac12)\) \(\approx\) \(2.032659278\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.58 + 1.58i)T \)
23 \( 1 + (-2.23 - 4.24i)T \)
good3 \( 1 - 2.23T + 3T^{2} \)
7 \( 1 - 2.82iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 5iT - 13T^{2} \)
17 \( 1 - 3.16T + 17T^{2} \)
19 \( 1 + 7.07T + 19T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 - 6.70iT - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + 1.41iT - 43T^{2} \)
47 \( 1 - 6.70T + 47T^{2} \)
53 \( 1 + 6.32T + 53T^{2} \)
59 \( 1 + 4.47iT - 59T^{2} \)
61 \( 1 - 9.48iT - 61T^{2} \)
67 \( 1 - 2.82iT - 67T^{2} \)
71 \( 1 - 6.70iT - 71T^{2} \)
73 \( 1 + 15iT - 73T^{2} \)
79 \( 1 - 7.07T + 79T^{2} \)
83 \( 1 - 12.7iT - 83T^{2} \)
89 \( 1 + 3.16iT - 89T^{2} \)
97 \( 1 - 18.9T + 97T^{2} \)
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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

−8.926579141241207677738711257564, −8.804959805782542760045329300788, −8.143852059933433271620037131537, −7.25264401365079436353428090041, −6.31574039558261127084812586185, −5.18163238952267419199409687311, −4.33724934602085685282624595351, −3.47650163852783045054048012208, −2.52938244745078273887913216445, −1.55067138006863016524635276581, 0.63577927884418886561814381951, 2.40339157054862129876239266136, 3.13293391683494459282705670544, 3.86883632478825642349661267485, 4.65028647184305883842286437200, 6.11794951832063837002994281039, 6.93368156049179411168850889910, 7.85858754952820194526814804888, 8.052470343578053694706244261137, 8.867369476501044784546700002439

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