Properties

Label 2-1840-5.4-c1-0-34
Degree $2$
Conductor $1840$
Sign $0.985 + 0.168i$
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  + 0.356i·3-s + (0.376 − 2.20i)5-s + 2.46i·7-s + 2.87·9-s − 1.61·11-s − 2.62i·13-s + (0.785 + 0.134i)15-s + 2.58i·17-s + 4.02·19-s − 0.878·21-s i·23-s + (−4.71 − 1.65i)25-s + 2.09i·27-s + 7.08·29-s + 4.58·31-s + ⋯
L(s)  = 1  + 0.205i·3-s + (0.168 − 0.985i)5-s + 0.931i·7-s + 0.957·9-s − 0.487·11-s − 0.728i·13-s + (0.202 + 0.0346i)15-s + 0.627i·17-s + 0.923·19-s − 0.191·21-s − 0.208i·23-s + (−0.943 − 0.331i)25-s + 0.402i·27-s + 1.31·29-s + 0.823·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.985 + 0.168i$
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: \((2,\ 1840,\ (\ :1/2),\ 0.985 + 0.168i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.945617418\)
\(L(\frac12)\) \(\approx\) \(1.945617418\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.376 + 2.20i)T \)
23 \( 1 + iT \)
good3 \( 1 - 0.356iT - 3T^{2} \)
7 \( 1 - 2.46iT - 7T^{2} \)
11 \( 1 + 1.61T + 11T^{2} \)
13 \( 1 + 2.62iT - 13T^{2} \)
17 \( 1 - 2.58iT - 17T^{2} \)
19 \( 1 - 4.02T + 19T^{2} \)
29 \( 1 - 7.08T + 29T^{2} \)
31 \( 1 - 4.58T + 31T^{2} \)
37 \( 1 + 2.96iT - 37T^{2} \)
41 \( 1 + 5.71T + 41T^{2} \)
43 \( 1 - 2.30iT - 43T^{2} \)
47 \( 1 - 6.88iT - 47T^{2} \)
53 \( 1 + 6.76iT - 53T^{2} \)
59 \( 1 - 2.53T + 59T^{2} \)
61 \( 1 - 9.25T + 61T^{2} \)
67 \( 1 + 15.7iT - 67T^{2} \)
71 \( 1 - 5.25T + 71T^{2} \)
73 \( 1 - 6.03iT - 73T^{2} \)
79 \( 1 + 1.35T + 79T^{2} \)
83 \( 1 - 8.59iT - 83T^{2} \)
89 \( 1 - 8.84T + 89T^{2} \)
97 \( 1 + 8.01iT - 97T^{2} \)
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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

−9.289286534754416873716341561222, −8.348813059503281253471584347745, −7.978751080362969553855633095730, −6.80400311800911836321388509134, −5.83036083446075592013654052219, −5.14183522991173194926363540234, −4.47854347349832476659860587417, −3.30850843939809449154352407676, −2.15837395442535619269949189105, −0.957199736531995539253762338435, 1.03639338196945115734507379872, 2.31372077556602765889268833238, 3.34700072338901225887664772828, 4.25906020581943831644939170828, 5.14764936510613672637445536639, 6.36447390762928507531299313130, 7.05044344605818692344243099668, 7.38241445499749635108663865194, 8.342436354518774161322259008485, 9.516603432748187574469965073750

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