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Properties

Label 2-1805-95.34-c0-0-0
Degree $2$
Conductor $1805$
Sign $-0.845 - 0.533i$
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  + (1.32 + 0.483i)2-s + (0.245 + 1.39i)3-s + (0.766 + 0.642i)4-s + (−0.766 + 0.642i)5-s + (−0.347 + 1.96i)6-s + (−0.939 + 0.342i)9-s + (−1.32 + 0.483i)10-s + (−0.707 + 1.22i)12-s + (−0.245 + 1.39i)13-s + (−1.08 − 0.909i)15-s + (−0.173 − 0.984i)16-s − 1.41·18-s − 0.999·20-s + (0.173 − 0.984i)25-s + (−1 + 1.73i)26-s + ⋯
L(s)  = 1  + (1.32 + 0.483i)2-s + (0.245 + 1.39i)3-s + (0.766 + 0.642i)4-s + (−0.766 + 0.642i)5-s + (−0.347 + 1.96i)6-s + (−0.939 + 0.342i)9-s + (−1.32 + 0.483i)10-s + (−0.707 + 1.22i)12-s + (−0.245 + 1.39i)13-s + (−1.08 − 0.909i)15-s + (−0.173 − 0.984i)16-s − 1.41·18-s − 0.999·20-s + (0.173 − 0.984i)25-s + (−1 + 1.73i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.845 - 0.533i$
Analytic conductor: \(0.900812\)
Root analytic conductor: \(0.949111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (984, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :0),\ -0.845 - 0.533i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.021686396\)
\(L(\frac12)\) \(\approx\) \(2.021686396\)
\(L(1)\) 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 \( 1 + (-1.32 - 0.483i)T + (0.766 + 0.642i)T^{2} \)
3 \( 1 + (-0.245 - 1.39i)T + (-0.939 + 0.342i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.245 - 1.39i)T + (-0.939 - 0.342i)T^{2} \)
17 \( 1 + (-0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.766 + 0.642i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - 1.41T + T^{2} \)
41 \( 1 + (0.939 - 0.342i)T^{2} \)
43 \( 1 + (-0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 + (-1.08 - 0.909i)T + (0.173 + 0.984i)T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (0.173 + 0.984i)T^{2} \)
67 \( 1 + (-1.32 + 0.483i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (0.939 - 0.342i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.939 + 0.342i)T^{2} \)
97 \( 1 + (1.32 + 0.483i)T + (0.766 + 0.642i)T^{2} \)
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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

−9.728282869934875049367839123695, −9.167823846669985766190971057867, −8.121133604253418552928820397039, −7.11029764811145618448815560641, −6.52884732589837190647599535818, −5.51012849723697429511291276726, −4.56631453167366523835540162179, −4.17334276673113207413300804850, −3.49443910345540789027740995870, −2.57663511492882243031011014158, 1.01337289331048794194138173222, 2.31253465616838711795928023632, 3.13234520492609238663944104419, 4.07753683118058383744688836439, 5.02716619181511411408551203972, 5.75152015933539729279390319539, 6.64744606856370262430806136757, 7.64676356087994395937613167583, 8.079055493071417520499133128539, 8.852190799626281882771096982399

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