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Properties

Label 2-1805-5.4-c1-0-76
Degree $2$
Conductor $1805$
Sign $-0.899 + 0.436i$
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  − 2.32i·2-s + 1.14i·3-s − 3.40·4-s + (−2.01 + 0.975i)5-s + 2.67·6-s − 0.143i·7-s + 3.26i·8-s + 1.68·9-s + (2.26 + 4.67i)10-s − 2.81·11-s − 3.90i·12-s + 1.70i·13-s − 0.333·14-s + (−1.12 − 2.31i)15-s + 0.778·16-s + 3.55i·17-s + ⋯
L(s)  = 1  − 1.64i·2-s + 0.663i·3-s − 1.70·4-s + (−0.899 + 0.436i)5-s + 1.09·6-s − 0.0541i·7-s + 1.15i·8-s + 0.560·9-s + (0.717 + 1.47i)10-s − 0.848·11-s − 1.12i·12-s + 0.473i·13-s − 0.0890·14-s + (−0.289 − 0.596i)15-s + 0.194·16-s + 0.863i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 + 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.899 + 0.436i$
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: \((2,\ 1805,\ (\ :1/2),\ -0.899 + 0.436i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9031050557\)
\(L(\frac12)\) \(\approx\) \(0.9031050557\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.01 - 0.975i)T \)
19 \( 1 \)
good2 \( 1 + 2.32iT - 2T^{2} \)
3 \( 1 - 1.14iT - 3T^{2} \)
7 \( 1 + 0.143iT - 7T^{2} \)
11 \( 1 + 2.81T + 11T^{2} \)
13 \( 1 - 1.70iT - 13T^{2} \)
17 \( 1 - 3.55iT - 17T^{2} \)
23 \( 1 + 7.19iT - 23T^{2} \)
29 \( 1 - 7.57T + 29T^{2} \)
31 \( 1 + 4.84T + 31T^{2} \)
37 \( 1 + 9.49iT - 37T^{2} \)
41 \( 1 - 0.187T + 41T^{2} \)
43 \( 1 + 10.9iT - 43T^{2} \)
47 \( 1 + 3.67iT - 47T^{2} \)
53 \( 1 - 1.64iT - 53T^{2} \)
59 \( 1 + 5.36T + 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 + 6.00iT - 67T^{2} \)
71 \( 1 + 0.540T + 71T^{2} \)
73 \( 1 + 10.4iT - 73T^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 + 7.59iT - 83T^{2} \)
89 \( 1 - 1.44T + 89T^{2} \)
97 \( 1 + 8.80iT - 97T^{2} \)
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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

−9.087541178664648100902627502394, −8.549637216720581931175754430219, −7.49272642192467112829198946789, −6.63063202961297388708713959519, −5.19373171963051548590792952123, −4.24996476685690754904981442778, −3.92334819633956788244100067051, −2.92264698898556075541517273450, −1.99037406716140122151757653296, −0.40921456267304949467872692028, 1.05969387098941588006476008630, 2.88238663271292945023740199533, 4.20955702151209546864566657801, 4.98809188850503008512448519557, 5.63901161654030945371869565906, 6.73625668973475952883757758198, 7.25197096094870622125589844670, 7.944137084371562656351701354755, 8.246994395434074589169970958604, 9.285115466194032393987968266074

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