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Properties

Label 2-1805-95.42-c0-0-0
Degree $2$
Conductor $1805$
Sign $-0.709 + 0.704i$
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 + (−1.36 + 0.366i)7-s + (−0.342 − 0.939i)9-s + (−0.173 − 0.984i)16-s + (−0.597 − 1.28i)17-s + i·20-s + (−1.40 + 0.123i)23-s + (0.173 − 0.984i)25-s + (−0.597 + 1.28i)28-s + (0.811 − 1.15i)35-s + (−0.939 − 0.342i)36-s + (0.123 − 1.40i)43-s + (0.866 + 0.5i)45-s + (−1.28 − 0.597i)47-s + ⋯
L(s)  = 1  + (0.642 − 0.766i)4-s + (−0.766 + 0.642i)5-s + (−1.36 + 0.366i)7-s + (−0.342 − 0.939i)9-s + (−0.173 − 0.984i)16-s + (−0.597 − 1.28i)17-s + i·20-s + (−1.40 + 0.123i)23-s + (0.173 − 0.984i)25-s + (−0.597 + 1.28i)28-s + (0.811 − 1.15i)35-s + (−0.939 − 0.342i)36-s + (0.123 − 1.40i)43-s + (0.866 + 0.5i)45-s + (−1.28 − 0.597i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.709 + 0.704i$
Analytic conductor: \(0.900812\)
Root analytic conductor: \(0.949111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (1182, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :0),\ -0.709 + 0.704i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5030846396\)
\(L(\frac12)\) \(\approx\) \(0.5030846396\)
\(L(1)\) 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 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.342 + 0.939i)T^{2} \)
17 \( 1 + (0.597 + 1.28i)T + (-0.642 + 0.766i)T^{2} \)
23 \( 1 + (1.40 - 0.123i)T + (0.984 - 0.173i)T^{2} \)
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 \( 1 + (-0.123 + 1.40i)T + (-0.984 - 0.173i)T^{2} \)
47 \( 1 + (1.28 + 0.597i)T + (0.642 + 0.766i)T^{2} \)
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 \( 1 + (-1.15 - 0.811i)T + (0.342 + 0.939i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
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

−9.440286355510871107092902891411, −8.433407425495988858717409555262, −7.28761811285294807981549047801, −6.67364822568785863254098024173, −6.22974114870588606817100308735, −5.30416186258105485323878057504, −3.93690940540000200548673935909, −3.12371103530665380156743823547, −2.33865530084199106035105353313, −0.34005321573832572035404083816, 1.91634048152943944396380247678, 3.09753847322977760940785880775, 3.84346253308110424663653513014, 4.62533877399846836748248385594, 6.00146535301962561152848881374, 6.56125734627026317341798071391, 7.60122289464399661086440974086, 8.071166358392073670213800357888, 8.754106516778917863821193659940, 9.767548246742453126574164379643

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