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Properties

Label 2-722-19.17-c1-0-24
Degree $2$
Conductor $722$
Sign $-0.999 - 0.00130i$
Analytic cond. $5.76519$
Root an. cond. $2.40108$
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  + (0.173 − 0.984i)2-s + (−0.766 − 0.642i)3-s + (−0.939 − 0.342i)4-s + (−0.766 + 0.642i)6-s + (0.5 + 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.347 − 1.96i)9-s + (3 − 5.19i)11-s + (0.499 + 0.866i)12-s + (−3.83 + 3.21i)13-s + (0.939 − 0.342i)14-s + (0.766 + 0.642i)16-s + (0.520 − 2.95i)17-s − 2·18-s + (0.173 − 0.984i)21-s + (−4.59 − 3.85i)22-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (−0.442 − 0.371i)3-s + (−0.469 − 0.171i)4-s + (−0.312 + 0.262i)6-s + (0.188 + 0.327i)7-s + (−0.176 + 0.306i)8-s + (−0.115 − 0.656i)9-s + (0.904 − 1.56i)11-s + (0.144 + 0.249i)12-s + (−1.06 + 0.891i)13-s + (0.251 − 0.0914i)14-s + (0.191 + 0.160i)16-s + (0.126 − 0.716i)17-s − 0.471·18-s + (0.0378 − 0.214i)21-s + (−0.979 − 0.822i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $-0.999 - 0.00130i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ -0.999 - 0.00130i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000563437 + 0.863237i\)
\(L(\frac12)\) \(\approx\) \(0.000563437 + 0.863237i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 \)
good3 \( 1 + (0.766 + 0.642i)T + (0.520 + 2.95i)T^{2} \)
5 \( 1 + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.83 - 3.21i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.520 + 2.95i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (2.81 + 1.02i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (1.56 + 8.86i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (7.51 - 2.73i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (2.81 + 1.02i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (1.56 - 8.86i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-9.39 - 3.42i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.868 + 4.92i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (5.63 - 2.05i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (5.36 + 4.49i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-7.66 - 6.42i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-9.19 + 7.71i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-1.73 + 9.84i)T + (-91.1 - 33.1i)T^{2} \)
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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

−9.879516702534192048712045899401, −9.295970474827427328869999781390, −8.472104186164830354342349319727, −7.28130542379918854667675523248, −6.22501423204417187553510043795, −5.58015507663847693662402630872, −4.27771942799594674446980480582, −3.30213451960574396325027067848, −1.94784069035861619578230033458, −0.44613405403961795590339865329, 1.93290542123712423715834431658, 3.71727111308198691168348134962, 4.73702389317742243238283135910, 5.29064296469945187160141977663, 6.46837549223015830317102397376, 7.37883270594260666638568526277, 7.982815750311210911127180134207, 9.156326307862592977516961638539, 10.11590045357002111390807117358, 10.49313989087027098783239062548

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