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Properties

Label 2-1445-5.4-c1-0-93
Degree $2$
Conductor $1445$
Sign $-0.931 + 0.363i$
Analytic cond. $11.5383$
Root an. cond. $3.39681$
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  − 0.232i·2-s − 2.39i·3-s + 1.94·4-s + (−2.08 + 0.812i)5-s − 0.556·6-s + 2.06i·7-s − 0.917i·8-s − 2.73·9-s + (0.188 + 0.484i)10-s − 0.480·11-s − 4.65i·12-s − 4.07i·13-s + 0.480·14-s + (1.94 + 4.98i)15-s + 3.67·16-s + ⋯
L(s)  = 1  − 0.164i·2-s − 1.38i·3-s + 0.972·4-s + (−0.931 + 0.363i)5-s − 0.227·6-s + 0.780i·7-s − 0.324i·8-s − 0.910·9-s + (0.0597 + 0.153i)10-s − 0.144·11-s − 1.34i·12-s − 1.12i·13-s + 0.128·14-s + (0.502 + 1.28i)15-s + 0.919·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 + 0.363i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.931 + 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1445\)    =    \(5 \cdot 17^{2}\)
Sign: $-0.931 + 0.363i$
Analytic conductor: \(11.5383\)
Root analytic conductor: \(3.39681\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1445} (579, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1445,\ (\ :1/2),\ -0.931 + 0.363i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.277883610\)
\(L(\frac12)\) \(\approx\) \(1.277883610\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.08 - 0.812i)T \)
17 \( 1 \)
good2 \( 1 + 0.232iT - 2T^{2} \)
3 \( 1 + 2.39iT - 3T^{2} \)
7 \( 1 - 2.06iT - 7T^{2} \)
11 \( 1 + 0.480T + 11T^{2} \)
13 \( 1 + 4.07iT - 13T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 8.15iT - 23T^{2} \)
29 \( 1 + 1.03T + 29T^{2} \)
31 \( 1 + 6.06T + 31T^{2} \)
37 \( 1 + 1.29iT - 37T^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 + 7.45iT - 43T^{2} \)
47 \( 1 - 3.60iT - 47T^{2} \)
53 \( 1 + 6.14iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 5.65T + 61T^{2} \)
67 \( 1 - 3.14iT - 67T^{2} \)
71 \( 1 - 1.81T + 71T^{2} \)
73 \( 1 + 12.1iT - 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + 2.23iT - 83T^{2} \)
89 \( 1 - 9.37T + 89T^{2} \)
97 \( 1 - 16.7iT - 97T^{2} \)
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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

−8.802507607056305964090129013090, −8.108318354451983389653284962425, −7.56640223887024180469460066139, −6.72327580109925425720536121529, −6.27596556235525121247755179771, −5.20831670355265096305318147483, −3.67129117369017194870810441125, −2.66545179539142377100737052359, −2.01060462159715679445105571729, −0.47872793324403517313010936962, 1.66851404209737456492844093561, 3.30653777291081007855059941483, 3.91267372278533789420123161916, 4.68672953656850780771085876170, 5.59655392615926168014757517220, 6.81479025899796054443989356678, 7.39174910457049577760376445109, 8.292785325359602537646841662433, 9.167114369374511390882752485613, 9.929920751729878781926316067016

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