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Properties

Label 2-1805-5.4-c1-0-153
Degree $2$
Conductor $1805$
Sign $0.765 + 0.643i$
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  − 2.68i·2-s − 2.14i·3-s − 5.21·4-s + (−1.71 − 1.43i)5-s − 5.76·6-s − 2.78i·7-s + 8.64i·8-s − 1.60·9-s + (−3.86 + 4.60i)10-s − 2.37·11-s + 11.2i·12-s + 0.0404i·13-s − 7.47·14-s + (−3.08 + 3.67i)15-s + 12.7·16-s + 1.81i·17-s + ⋯
L(s)  = 1  − 1.89i·2-s − 1.23i·3-s − 2.60·4-s + (−0.765 − 0.643i)5-s − 2.35·6-s − 1.05i·7-s + 3.05i·8-s − 0.535·9-s + (−1.22 + 1.45i)10-s − 0.717·11-s + 3.23i·12-s + 0.0112i·13-s − 1.99·14-s + (−0.797 + 0.949i)15-s + 3.19·16-s + 0.440i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.765 + 0.643i$
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: \((2,\ 1805,\ (\ :1/2),\ 0.765 + 0.643i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3560113633\)
\(L(\frac12)\) \(\approx\) \(0.3560113633\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.71 + 1.43i)T \)
19 \( 1 \)
good2 \( 1 + 2.68iT - 2T^{2} \)
3 \( 1 + 2.14iT - 3T^{2} \)
7 \( 1 + 2.78iT - 7T^{2} \)
11 \( 1 + 2.37T + 11T^{2} \)
13 \( 1 - 0.0404iT - 13T^{2} \)
17 \( 1 - 1.81iT - 17T^{2} \)
23 \( 1 + 2.54iT - 23T^{2} \)
29 \( 1 + 2.94T + 29T^{2} \)
31 \( 1 + 2.88T + 31T^{2} \)
37 \( 1 - 0.227iT - 37T^{2} \)
41 \( 1 - 8.03T + 41T^{2} \)
43 \( 1 - 5.13iT - 43T^{2} \)
47 \( 1 + 11.0iT - 47T^{2} \)
53 \( 1 - 5.71iT - 53T^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 - 4.58T + 61T^{2} \)
67 \( 1 + 4.85iT - 67T^{2} \)
71 \( 1 + 7.76T + 71T^{2} \)
73 \( 1 + 4.24iT - 73T^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 - 4.47iT - 83T^{2} \)
89 \( 1 + 15.3T + 89T^{2} \)
97 \( 1 + 5.59iT - 97T^{2} \)
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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

−8.447434507891262430393625580281, −7.81253931626725609877885260639, −7.16437552084827602718266341966, −5.73766511572662173296640039396, −4.60341898615512898256374765319, −4.03237968838735676845090211224, −3.03550925034925456087272665113, −1.91149757956079153544693873864, −1.03923202371705464541365070169, −0.16415434515187221894882204158, 2.92906858557055432285193683775, 3.92394532324875475626144411435, 4.64973143740459415441430424157, 5.44752060443012362208498343003, 5.99471434486182000502003090319, 7.13439147417205868018170885385, 7.64824490550032745499044801584, 8.500141747935860161516188205688, 9.163674111962691391542675444319, 9.738346355338556680541248573022

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