Properties

Label 2-1805-95.23-c0-0-0
Degree $2$
Conductor $1805$
Sign $-0.947 - 0.319i$
Analytic cond. $0.900812$
Root an. cond. $0.949111$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.642 + 0.766i)4-s + (−0.766 + 0.642i)5-s + (0.366 + 1.36i)7-s + (0.342 + 0.939i)9-s + (−0.173 − 0.984i)16-s + (−1.28 + 0.597i)17-s i·20-s + (−0.123 − 1.40i)23-s + (0.173 − 0.984i)25-s + (−1.28 − 0.597i)28-s + (−1.15 − 0.811i)35-s + (−0.939 − 0.342i)36-s + (1.40 + 0.123i)43-s + (−0.866 − 0.5i)45-s + (−0.597 + 1.28i)47-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)4-s + (−0.766 + 0.642i)5-s + (0.366 + 1.36i)7-s + (0.342 + 0.939i)9-s + (−0.173 − 0.984i)16-s + (−1.28 + 0.597i)17-s i·20-s + (−0.123 − 1.40i)23-s + (0.173 − 0.984i)25-s + (−1.28 − 0.597i)28-s + (−1.15 − 0.811i)35-s + (−0.939 − 0.342i)36-s + (1.40 + 0.123i)43-s + (−0.866 − 0.5i)45-s + (−0.597 + 1.28i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.947 - 0.319i$
Analytic conductor: \(0.900812\)
Root analytic conductor: \(0.949111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (1543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :0),\ -0.947 - 0.319i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6731099790\)
\(L(\frac12)\) \(\approx\) \(0.6731099790\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 \)
good2 \( 1 + (0.642 - 0.766i)T^{2} \)
3 \( 1 + (-0.342 - 0.939i)T^{2} \)
7 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.342 - 0.939i)T^{2} \)
17 \( 1 + (1.28 - 0.597i)T + (0.642 - 0.766i)T^{2} \)
23 \( 1 + (0.123 + 1.40i)T + (-0.984 + 0.173i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (-1.40 - 0.123i)T + (0.984 + 0.173i)T^{2} \)
47 \( 1 + (0.597 - 1.28i)T + (-0.642 - 0.766i)T^{2} \)
53 \( 1 + (-0.984 + 0.173i)T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (0.173 + 0.984i)T^{2} \)
67 \( 1 + (-0.642 - 0.766i)T^{2} \)
71 \( 1 + (0.173 - 0.984i)T^{2} \)
73 \( 1 + (0.811 - 1.15i)T + (-0.342 - 0.939i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
89 \( 1 + (0.939 + 0.342i)T^{2} \)
97 \( 1 + (0.642 - 0.766i)T^{2} \)
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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

−9.690700320658930370831366307887, −8.663378704748684819405902489025, −8.403636700280602094215361553520, −7.63054230926474926306961030031, −6.77502498644918294418496675944, −5.76965296068016682937064694560, −4.62252078876531456656121165923, −4.23424579305090711524615864850, −2.88939666291090181038396089750, −2.24014108874784690126845988077, 0.53367304276813458563710460650, 1.54445756601223157737288484673, 3.55216028877336498766659990663, 4.23580961639234187909216113017, 4.76446943483780842279081797165, 5.79346642440716879587197869214, 6.90650403208475704519850871499, 7.44145067060815182399686487604, 8.449145933388672426013932340959, 9.182182832435953302260959176544

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