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Properties

Label 2-1445-5.4-c1-0-53
Degree $2$
Conductor $1445$
Sign $0.0796 - 0.996i$
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  + 1.80i·2-s − 0.561i·3-s − 1.25·4-s + (0.178 − 2.22i)5-s + 1.01·6-s + 3.11i·7-s + 1.34i·8-s + 2.68·9-s + (4.01 + 0.321i)10-s + 5.61·11-s + 0.703i·12-s − 1.24i·13-s − 5.61·14-s + (−1.25 − 0.100i)15-s − 4.93·16-s + ⋯
L(s)  = 1  + 1.27i·2-s − 0.324i·3-s − 0.626·4-s + (0.0796 − 0.996i)5-s + 0.413·6-s + 1.17i·7-s + 0.476i·8-s + 0.894·9-s + (1.27 + 0.101i)10-s + 1.69·11-s + 0.203i·12-s − 0.346i·13-s − 1.49·14-s + (−0.323 − 0.0258i)15-s − 1.23·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0796 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0796 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1445\)    =    \(5 \cdot 17^{2}\)
Sign: $0.0796 - 0.996i$
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: \((2,\ 1445,\ (\ :1/2),\ 0.0796 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.094121664\)
\(L(\frac12)\) \(\approx\) \(2.094121664\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.178 + 2.22i)T \)
17 \( 1 \)
good2 \( 1 - 1.80iT - 2T^{2} \)
3 \( 1 + 0.561iT - 3T^{2} \)
7 \( 1 - 3.11iT - 7T^{2} \)
11 \( 1 - 5.61T + 11T^{2} \)
13 \( 1 + 1.24iT - 13T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 2.32iT - 23T^{2} \)
29 \( 1 - 6.62T + 29T^{2} \)
31 \( 1 - 4.55T + 31T^{2} \)
37 \( 1 - 1.90iT - 37T^{2} \)
41 \( 1 - 5.92T + 41T^{2} \)
43 \( 1 + 2.04iT - 43T^{2} \)
47 \( 1 - 4.85iT - 47T^{2} \)
53 \( 1 - 9.11iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 5.65T + 61T^{2} \)
67 \( 1 - 8.46iT - 67T^{2} \)
71 \( 1 + 8.79T + 71T^{2} \)
73 \( 1 - 1.56iT - 73T^{2} \)
79 \( 1 + 4.91T + 79T^{2} \)
83 \( 1 + 3.94iT - 83T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 + 12.3iT - 97T^{2} \)
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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

−9.212628131510390724761480617492, −8.765051028346086558269146991044, −8.154835812460641944604232770760, −7.20501741500977452942458377601, −6.26248373426958390345212444944, −6.01756025923642748296664795802, −4.79252596947461487530513698523, −4.24079418450321334765332286245, −2.43116980635679018404134992694, −1.25071451876621200374146399375, 1.04940620802546342574655172915, 2.02930362778332854636379122518, 3.33378409688452242478373795601, 4.00331426845810679423517450880, 4.47716269252410670676391617901, 6.47033511266062858216457511780, 6.71148967542710863736343278666, 7.58741347106718714168213786347, 8.962169286096430080025993633124, 9.756419001771142555887772462354

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