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Properties

Label 2-1805-5.4-c1-0-41
Degree $2$
Conductor $1805$
Sign $-0.378 - 0.925i$
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  + 1.40i·2-s − 2.85i·3-s + 0.0388·4-s + (0.846 + 2.06i)5-s + 3.99·6-s + 4.34i·7-s + 2.85i·8-s − 5.13·9-s + (−2.89 + 1.18i)10-s + 3.16·11-s − 0.110i·12-s − 2.49i·13-s − 6.08·14-s + (5.90 − 2.41i)15-s − 3.92·16-s + 2.87i·17-s + ⋯
L(s)  = 1  + 0.990i·2-s − 1.64i·3-s + 0.0194·4-s + (0.378 + 0.925i)5-s + 1.63·6-s + 1.64i·7-s + 1.00i·8-s − 1.71·9-s + (−0.916 + 0.374i)10-s + 0.954·11-s − 0.0320i·12-s − 0.692i·13-s − 1.62·14-s + (1.52 − 0.623i)15-s − 0.980·16-s + 0.696i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.378 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.378 - 0.925i$
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: \((2,\ 1805,\ (\ :1/2),\ -0.378 - 0.925i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.882035684\)
\(L(\frac12)\) \(\approx\) \(1.882035684\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.846 - 2.06i)T \)
19 \( 1 \)
good2 \( 1 - 1.40iT - 2T^{2} \)
3 \( 1 + 2.85iT - 3T^{2} \)
7 \( 1 - 4.34iT - 7T^{2} \)
11 \( 1 - 3.16T + 11T^{2} \)
13 \( 1 + 2.49iT - 13T^{2} \)
17 \( 1 - 2.87iT - 17T^{2} \)
23 \( 1 + 0.227iT - 23T^{2} \)
29 \( 1 + 1.91T + 29T^{2} \)
31 \( 1 + 4.19T + 31T^{2} \)
37 \( 1 - 8.87iT - 37T^{2} \)
41 \( 1 + 4.01T + 41T^{2} \)
43 \( 1 - 4.41iT - 43T^{2} \)
47 \( 1 + 12.3iT - 47T^{2} \)
53 \( 1 - 6.04iT - 53T^{2} \)
59 \( 1 - 2.58T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 + 3.61iT - 67T^{2} \)
71 \( 1 + 4.59T + 71T^{2} \)
73 \( 1 - 2.68iT - 73T^{2} \)
79 \( 1 - 6.63T + 79T^{2} \)
83 \( 1 + 1.01iT - 83T^{2} \)
89 \( 1 - 0.286T + 89T^{2} \)
97 \( 1 + 8.11iT - 97T^{2} \)
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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

−9.056740626807061978508742689404, −8.416093222184516409758121017664, −7.79890955787016364699179098540, −6.91568572027211520367268502257, −6.46350955071000153335187385809, −5.88496708932805573531886125314, −5.32346761035477011749385718236, −3.25893486286738688437039381471, −2.33566281601172497088154353447, −1.70899134239207550778035120305, 0.66969059718055533219782215253, 1.86924301012655419064573075090, 3.37028912868624766109213500064, 4.06800626111539280865936555749, 4.39544599700569179999025228336, 5.43228613097029666952084851681, 6.61805707776046236594867669826, 7.44635182848765846686759368995, 8.795832732027420442832990284331, 9.420912625460257233759261234477

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