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Properties

Label 2-722-19.5-c1-0-4
Degree $2$
Conductor $722$
Sign $-0.0534 - 0.998i$
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  + (−0.939 + 0.342i)2-s + (0.0603 − 0.342i)3-s + (0.766 − 0.642i)4-s + (1.53 + 1.28i)5-s + (0.0603 + 0.342i)6-s + (0.879 + 1.52i)7-s + (−0.500 + 0.866i)8-s + (2.70 + 0.984i)9-s + (−1.87 − 0.684i)10-s + (−2.11 + 3.66i)11-s + (−0.173 − 0.300i)12-s + (0.184 + 1.04i)13-s + (−1.34 − 1.13i)14-s + (0.532 − 0.446i)15-s + (0.173 − 0.984i)16-s + (−6.69 + 2.43i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (0.0348 − 0.197i)3-s + (0.383 − 0.321i)4-s + (0.685 + 0.574i)5-s + (0.0246 + 0.139i)6-s + (0.332 + 0.575i)7-s + (−0.176 + 0.306i)8-s + (0.901 + 0.328i)9-s + (−0.594 − 0.216i)10-s + (−0.637 + 1.10i)11-s + (−0.0501 − 0.0868i)12-s + (0.0512 + 0.290i)13-s + (−0.360 − 0.302i)14-s + (0.137 − 0.115i)15-s + (0.0434 − 0.246i)16-s + (−1.62 + 0.591i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0534 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0534 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $-0.0534 - 0.998i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ -0.0534 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.821635 + 0.866789i\)
\(L(\frac12)\) \(\approx\) \(0.821635 + 0.866789i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 \)
good3 \( 1 + (-0.0603 + 0.342i)T + (-2.81 - 1.02i)T^{2} \)
5 \( 1 + (-1.53 - 1.28i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-0.879 - 1.52i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.11 - 3.66i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.184 - 1.04i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (6.69 - 2.43i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-2.87 + 2.41i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (5.98 + 2.17i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-4.41 - 7.64i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.45T + 37T^{2} \)
41 \( 1 + (-0.326 + 1.85i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-2.83 - 2.37i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-3.45 - 1.25i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (7.57 - 6.35i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-6.85 + 2.49i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-4.06 + 3.41i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-10.9 - 3.98i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-2.16 - 1.81i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-0.137 + 0.780i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (1.16 - 6.59i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (0.754 + 1.30i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.06 - 11.7i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (1.76 - 0.642i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.49359045198842431998087196528, −9.810349266891381674220723710470, −8.925292750647151125296211019132, −8.074815124809273358047682409126, −6.98245896754663951371453807090, −6.61459809733432029235543431134, −5.34309119980350395943933164083, −4.37497030464239569246399901584, −2.42440842108811029778468413040, −1.84752818571102867198932567584, 0.76495207097521342370367671049, 2.08933183538911546845797259667, 3.52375857245775202365666341109, 4.68150884345896726489905263866, 5.69963337123082062422865135333, 6.85337405558913180344681040331, 7.67469373564659804630951984161, 8.719362090317616046654098527767, 9.309965843478943958482356880180, 10.11106507086471598394293156498

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