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Properties

Label 2-1445-17.16-c1-0-11
Degree $2$
Conductor $1445$
Sign $-0.685 - 0.727i$
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  + 0.783·2-s + 0.544i·3-s − 1.38·4-s + i·5-s + 0.426i·6-s − 1.18i·7-s − 2.65·8-s + 2.70·9-s + 0.783i·10-s + 2.55i·11-s − 0.755i·12-s + 0.368·13-s − 0.931i·14-s − 0.544·15-s + 0.693·16-s + ⋯
L(s)  = 1  + 0.554·2-s + 0.314i·3-s − 0.693·4-s + 0.447i·5-s + 0.174i·6-s − 0.449i·7-s − 0.937·8-s + 0.901·9-s + 0.247i·10-s + 0.770i·11-s − 0.218i·12-s + 0.102·13-s − 0.248i·14-s − 0.140·15-s + 0.173·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 - 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1445\)    =    \(5 \cdot 17^{2}\)
Sign: $-0.685 - 0.727i$
Analytic conductor: \(11.5383\)
Root analytic conductor: \(3.39681\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1445} (866, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1445,\ (\ :1/2),\ -0.685 - 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.127169495\)
\(L(\frac12)\) \(\approx\) \(1.127169495\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - iT \)
17 \( 1 \)
good2 \( 1 - 0.783T + 2T^{2} \)
3 \( 1 - 0.544iT - 3T^{2} \)
7 \( 1 + 1.18iT - 7T^{2} \)
11 \( 1 - 2.55iT - 11T^{2} \)
13 \( 1 - 0.368T + 13T^{2} \)
19 \( 1 + 6.61T + 19T^{2} \)
23 \( 1 - 3.86iT - 23T^{2} \)
29 \( 1 - 2.31iT - 29T^{2} \)
31 \( 1 - 6.62iT - 31T^{2} \)
37 \( 1 - 3.17iT - 37T^{2} \)
41 \( 1 + 7.30iT - 41T^{2} \)
43 \( 1 + 6.82T + 43T^{2} \)
47 \( 1 + 7.80T + 47T^{2} \)
53 \( 1 + 8.01T + 53T^{2} \)
59 \( 1 + 5.22T + 59T^{2} \)
61 \( 1 - 8.12iT - 61T^{2} \)
67 \( 1 + 7.94T + 67T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 - 14.7iT - 73T^{2} \)
79 \( 1 + 0.813iT - 79T^{2} \)
83 \( 1 + 3.99T + 83T^{2} \)
89 \( 1 + 9.14T + 89T^{2} \)
97 \( 1 - 7.06iT - 97T^{2} \)
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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

−10.01127044293196854761101536840, −9.088322604804690523970771571417, −8.302578844820390342839770606940, −7.19977184801191285390804712694, −6.62635301118760040754233632335, −5.48847138854506172372247384546, −4.58143758145832021211070452464, −4.05287076820741966780665232529, −3.13439665797165057830658133416, −1.62227292616288565268708111187, 0.38373728008802148835521640814, 1.93372773280922176716011562389, 3.25550405229871162994044800089, 4.31997738443418470956530484134, 4.81010923488471701119193004347, 6.03809044257742858157833659838, 6.39423711326786161631666006318, 7.82195761822828165957239480836, 8.425036210515819691578153004695, 9.152298219244336901324930633833

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