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Properties

Label 2-1805-5.4-c1-0-135
Degree $2$
Conductor $1805$
Sign $-0.362 + 0.931i$
Analytic cond. $14.4129$
Root an. cond. $3.79644$
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  + 0.717i·2-s − 1.56i·3-s + 1.48·4-s + (0.811 − 2.08i)5-s + 1.12·6-s − 2.51i·7-s + 2.49i·8-s + 0.535·9-s + (1.49 + 0.581i)10-s − 5.85·11-s − 2.33i·12-s + 0.791i·13-s + 1.80·14-s + (−3.27 − 1.27i)15-s + 1.17·16-s − 0.651i·17-s + ⋯
L(s)  = 1  + 0.507i·2-s − 0.906i·3-s + 0.742·4-s + (0.362 − 0.931i)5-s + 0.459·6-s − 0.952i·7-s + 0.883i·8-s + 0.178·9-s + (0.472 + 0.183i)10-s − 1.76·11-s − 0.673i·12-s + 0.219i·13-s + 0.482·14-s + (−0.844 − 0.328i)15-s + 0.294·16-s − 0.157i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.362 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.362 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.362 + 0.931i$
Analytic conductor: \(14.4129\)
Root analytic conductor: \(3.79644\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (1084, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :1/2),\ -0.362 + 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.893035836\)
\(L(\frac12)\) \(\approx\) \(1.893035836\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.811 + 2.08i)T \)
19 \( 1 \)
good2 \( 1 - 0.717iT - 2T^{2} \)
3 \( 1 + 1.56iT - 3T^{2} \)
7 \( 1 + 2.51iT - 7T^{2} \)
11 \( 1 + 5.85T + 11T^{2} \)
13 \( 1 - 0.791iT - 13T^{2} \)
17 \( 1 + 0.651iT - 17T^{2} \)
23 \( 1 + 4.88iT - 23T^{2} \)
29 \( 1 - 4.83T + 29T^{2} \)
31 \( 1 + 6.73T + 31T^{2} \)
37 \( 1 - 0.741iT - 37T^{2} \)
41 \( 1 - 8.04T + 41T^{2} \)
43 \( 1 + 0.761iT - 43T^{2} \)
47 \( 1 + 11.3iT - 47T^{2} \)
53 \( 1 + 12.8iT - 53T^{2} \)
59 \( 1 - 2.14T + 59T^{2} \)
61 \( 1 + 6.75T + 61T^{2} \)
67 \( 1 - 13.8iT - 67T^{2} \)
71 \( 1 + 6.05T + 71T^{2} \)
73 \( 1 - 11.1iT - 73T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 - 3.26iT - 83T^{2} \)
89 \( 1 + 1.07T + 89T^{2} \)
97 \( 1 - 10.1iT - 97T^{2} \)
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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

−8.599494151763973106553039321050, −8.092719232318750984499134091100, −7.30753171776253158882883231748, −6.90780102030937459222925340377, −5.88420863935429351533940967407, −5.17905788911828694964542101261, −4.23619375725826418104195089845, −2.66119348978506415817527992020, −1.85371038217531286505740510647, −0.64021340500570327331156738583, 1.82825922047533352511154062269, 2.80216709507412754574118848282, 3.22907702468402271617562897350, 4.51538193531683274855819978855, 5.62668841997583907413619176979, 6.03567102046937316709928225423, 7.34432979042380859020786193944, 7.71372419861061978021781073676, 9.095066961765147994704579651548, 9.709470395336586540195600985292

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