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Properties

Label 2-1805-5.4-c1-0-47
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

\(L(1)\) \(\approx\) \(0.9592717698\)
\(L(\frac12)\) \(\approx\) \(0.9592717698\)
\(L(\frac{3}{2})\) 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

−9.346839741500337395859017225849, −8.453036097439031854611113118443, −8.106175389666730203719025701593, −7.20908874164811862686978388292, −6.53563033025188491294737732421, −5.24018716436499503365956507289, −4.92272301652379021168310869515, −4.07681721146365680050034654291, −3.60558266668278161944348242766, −0.59425222747321541648348001276, 0.74366983614690992027864065850, 2.05576784483752212871432795046, 2.64618492270067935714482740227, 3.43272745739682511045408486128, 4.59485511360830527774037325451, 5.52450122844924790183720703188, 6.64399424703446012839860706254, 7.70479745546983021704288022946, 8.273422876294270748008690232791, 9.046994500391980203434363421715

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