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Properties

Label 2-1805-5.4-c1-0-70
Degree $2$
Conductor $1805$
Sign $-0.944 - 0.328i$
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  + 1.47i·2-s + 2.52i·3-s − 0.177·4-s + (2.11 + 0.735i)5-s − 3.72·6-s − 4.13i·7-s + 2.68i·8-s − 3.38·9-s + (−1.08 + 3.11i)10-s + 5.45·11-s − 0.449i·12-s − 1.01i·13-s + 6.09·14-s + (−1.85 + 5.33i)15-s − 4.32·16-s + 1.45i·17-s + ⋯
L(s)  = 1  + 1.04i·2-s + 1.45i·3-s − 0.0889·4-s + (0.944 + 0.328i)5-s − 1.52·6-s − 1.56i·7-s + 0.950i·8-s − 1.12·9-s + (−0.343 + 0.985i)10-s + 1.64·11-s − 0.129i·12-s − 0.280i·13-s + 1.62·14-s + (−0.479 + 1.37i)15-s − 1.08·16-s + 0.352i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.328i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.944 - 0.328i$
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: \((2,\ 1805,\ (\ :1/2),\ -0.944 - 0.328i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.533914502\)
\(L(\frac12)\) \(\approx\) \(2.533914502\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.11 - 0.735i)T \)
19 \( 1 \)
good2 \( 1 - 1.47iT - 2T^{2} \)
3 \( 1 - 2.52iT - 3T^{2} \)
7 \( 1 + 4.13iT - 7T^{2} \)
11 \( 1 - 5.45T + 11T^{2} \)
13 \( 1 + 1.01iT - 13T^{2} \)
17 \( 1 - 1.45iT - 17T^{2} \)
23 \( 1 - 0.652iT - 23T^{2} \)
29 \( 1 - 7.82T + 29T^{2} \)
31 \( 1 + 7.28T + 31T^{2} \)
37 \( 1 + 2.71iT - 37T^{2} \)
41 \( 1 + 4.35T + 41T^{2} \)
43 \( 1 - 7.14iT - 43T^{2} \)
47 \( 1 - 1.47iT - 47T^{2} \)
53 \( 1 - 3.40iT - 53T^{2} \)
59 \( 1 + 9.52T + 59T^{2} \)
61 \( 1 + 0.766T + 61T^{2} \)
67 \( 1 + 2.01iT - 67T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 - 4.69iT - 73T^{2} \)
79 \( 1 - 1.89T + 79T^{2} \)
83 \( 1 - 9.50iT - 83T^{2} \)
89 \( 1 - 5.90T + 89T^{2} \)
97 \( 1 + 11.5iT - 97T^{2} \)
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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

−9.546707488259495632328904175861, −9.004521822423111877665488547256, −7.987667392204324713270562774968, −6.97666587698241579129390389554, −6.52635821224362317091802673728, −5.67429656464343050249880551675, −4.74518969940937961909091380327, −4.03286951546155083219047486515, −3.14546099942404833723773138689, −1.49365553097270101491041635358, 1.03390223353357765894388865540, 1.88265352589101614011515483881, 2.32753079687276830962654167600, 3.40905057642833663082127855735, 4.86559039495173704261748681798, 6.00597651567538051157737282319, 6.46248933357334757179982227720, 7.08559364073024185166118429545, 8.409163076462695911531874362829, 9.064912171248140708302768256875

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