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Properties

Label 2-1805-5.4-c1-0-122
Degree $2$
Conductor $1805$
Sign $0.315 + 0.948i$
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  + 2.66i·2-s − 1.76i·3-s − 5.12·4-s + (−0.705 − 2.12i)5-s + 4.70·6-s − 2.20i·7-s − 8.33i·8-s − 0.106·9-s + (5.66 − 1.88i)10-s + 2.86·11-s + 9.02i·12-s − 4.83i·13-s + 5.89·14-s + (−3.73 + 1.24i)15-s + 12.0·16-s + 2.06i·17-s + ⋯
L(s)  = 1  + 1.88i·2-s − 1.01i·3-s − 2.56·4-s + (−0.315 − 0.948i)5-s + 1.92·6-s − 0.835i·7-s − 2.94i·8-s − 0.0353·9-s + (1.79 − 0.595i)10-s + 0.864·11-s + 2.60i·12-s − 1.34i·13-s + 1.57·14-s + (−0.965 + 0.320i)15-s + 3.00·16-s + 0.500i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.315 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.315 + 0.948i$
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: \((2,\ 1805,\ (\ :1/2),\ 0.315 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9826591172\)
\(L(\frac12)\) \(\approx\) \(0.9826591172\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.705 + 2.12i)T \)
19 \( 1 \)
good2 \( 1 - 2.66iT - 2T^{2} \)
3 \( 1 + 1.76iT - 3T^{2} \)
7 \( 1 + 2.20iT - 7T^{2} \)
11 \( 1 - 2.86T + 11T^{2} \)
13 \( 1 + 4.83iT - 13T^{2} \)
17 \( 1 - 2.06iT - 17T^{2} \)
23 \( 1 - 4.40iT - 23T^{2} \)
29 \( 1 + 8.08T + 29T^{2} \)
31 \( 1 - 3.97T + 31T^{2} \)
37 \( 1 + 9.05iT - 37T^{2} \)
41 \( 1 - 1.21T + 41T^{2} \)
43 \( 1 + 5.13iT - 43T^{2} \)
47 \( 1 + 2.50iT - 47T^{2} \)
53 \( 1 - 1.22iT - 53T^{2} \)
59 \( 1 + 0.244T + 59T^{2} \)
61 \( 1 - 0.498T + 61T^{2} \)
67 \( 1 - 7.09iT - 67T^{2} \)
71 \( 1 + 9.59T + 71T^{2} \)
73 \( 1 - 0.287iT - 73T^{2} \)
79 \( 1 + 15.8T + 79T^{2} \)
83 \( 1 - 5.70iT - 83T^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 - 0.202iT - 97T^{2} \)
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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

−8.684562595748875862615615601565, −8.020341393585862901526578594436, −7.37264320447164209579298164173, −7.07585604723177790479256314448, −5.89973524851815753280564818739, −5.51607949266110008534334140120, −4.29659297654248935912971132927, −3.76959069522357428863082646187, −1.36088834668849364272765663512, −0.41186798242859900926790881293, 1.62513388394692007972767406479, 2.63834372736775155955754113851, 3.42066738665876274920499962241, 4.26068938286173328592343737114, 4.68942560884531212598453210813, 5.98829930466265355898226562799, 7.06523674097465591827384481016, 8.425728125178733120491868249269, 9.149805712578580509027632007802, 9.614271402621535351119734742209

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