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Properties

Label 2-1840-460.459-c1-0-9
Degree $2$
Conductor $1840$
Sign $-0.999 - 0.0241i$
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  + 1.49·3-s + (−0.689 + 2.12i)5-s + 4.18i·7-s − 0.754·9-s − 6.19·11-s − 0.773i·13-s + (−1.03 + 3.18i)15-s − 0.0856·17-s + 4.78·19-s + 6.27i·21-s + (3.29 − 3.48i)23-s + (−4.04 − 2.93i)25-s − 5.62·27-s + 6.23·29-s − 1.76i·31-s + ⋯
L(s)  = 1  + 0.865·3-s + (−0.308 + 0.951i)5-s + 1.58i·7-s − 0.251·9-s − 1.86·11-s − 0.214i·13-s + (−0.266 + 0.823i)15-s − 0.0207·17-s + 1.09·19-s + 1.36i·21-s + (0.687 − 0.726i)23-s + (−0.809 − 0.586i)25-s − 1.08·27-s + 1.15·29-s − 0.316i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.999 - 0.0241i$
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: \((2,\ 1840,\ (\ :1/2),\ -0.999 - 0.0241i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9001103792\)
\(L(\frac12)\) \(\approx\) \(0.9001103792\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.689 - 2.12i)T \)
23 \( 1 + (-3.29 + 3.48i)T \)
good3 \( 1 - 1.49T + 3T^{2} \)
7 \( 1 - 4.18iT - 7T^{2} \)
11 \( 1 + 6.19T + 11T^{2} \)
13 \( 1 + 0.773iT - 13T^{2} \)
17 \( 1 + 0.0856T + 17T^{2} \)
19 \( 1 - 4.78T + 19T^{2} \)
29 \( 1 - 6.23T + 29T^{2} \)
31 \( 1 + 1.76iT - 31T^{2} \)
37 \( 1 + 6.81T + 37T^{2} \)
41 \( 1 + 6.59T + 41T^{2} \)
43 \( 1 - 1.54iT - 43T^{2} \)
47 \( 1 + 9.34T + 47T^{2} \)
53 \( 1 + 8.12T + 53T^{2} \)
59 \( 1 - 12.7iT - 59T^{2} \)
61 \( 1 - 0.753iT - 61T^{2} \)
67 \( 1 - 3.43iT - 67T^{2} \)
71 \( 1 - 5.70iT - 71T^{2} \)
73 \( 1 + 11.8iT - 73T^{2} \)
79 \( 1 - 12.2T + 79T^{2} \)
83 \( 1 - 14.0iT - 83T^{2} \)
89 \( 1 + 0.211iT - 89T^{2} \)
97 \( 1 - 5.35T + 97T^{2} \)
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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

−9.586133295794618635714584311889, −8.634300355584987397192165742701, −8.183809657157141764680080295091, −7.52356740865584506453296142063, −6.48385692665393468160326608088, −5.54645214688551756923646217259, −4.93714380804964810858355772055, −3.23794923070364480591583371021, −2.87443170635691384263504917809, −2.22129152405362075291717809029, 0.27477880634957449161695165740, 1.60385697694976862235999089926, 3.05809093570826869437589464418, 3.62607764075743242805878910627, 4.89863981010868426458866088569, 5.20481460194707149466533972036, 6.71765332545235348239475746553, 7.73621748352488619863613229872, 7.88024657087687010034832898412, 8.729096430396320825040396173361

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