Properties

Label 2-644-161.103-c1-0-12
Degree $2$
Conductor $644$
Sign $-0.725 + 0.688i$
Analytic cond. $5.14236$
Root an. cond. $2.26767$
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  + (−0.548 − 1.37i)3-s + (0.0440 − 0.181i)5-s + (1.60 − 2.10i)7-s + (0.593 − 0.565i)9-s + (−5.75 − 0.273i)11-s + (−0.543 − 0.471i)13-s + (−0.273 + 0.0393i)15-s + (2.18 − 3.06i)17-s + (−0.00719 − 0.0100i)19-s + (−3.76 − 1.04i)21-s + (−0.696 + 4.74i)23-s + (4.41 + 2.27i)25-s + (−5.13 − 2.34i)27-s + (−3.60 − 7.89i)29-s + (−1.42 − 1.81i)31-s + ⋯
L(s)  = 1  + (−0.316 − 0.791i)3-s + (0.0197 − 0.0812i)5-s + (0.607 − 0.794i)7-s + (0.197 − 0.188i)9-s + (−1.73 − 0.0826i)11-s + (−0.150 − 0.130i)13-s + (−0.0705 + 0.0101i)15-s + (0.529 − 0.744i)17-s + (−0.00164 − 0.00231i)19-s + (−0.821 − 0.228i)21-s + (−0.145 + 0.989i)23-s + (0.882 + 0.455i)25-s + (−0.987 − 0.450i)27-s + (−0.669 − 1.46i)29-s + (−0.256 − 0.326i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.725 + 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.725 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $-0.725 + 0.688i$
Analytic conductor: \(5.14236\)
Root analytic conductor: \(2.26767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :1/2),\ -0.725 + 0.688i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.396120 - 0.993276i\)
\(L(\frac12)\) \(\approx\) \(0.396120 - 0.993276i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-1.60 + 2.10i)T \)
23 \( 1 + (0.696 - 4.74i)T \)
good3 \( 1 + (0.548 + 1.37i)T + (-2.17 + 2.07i)T^{2} \)
5 \( 1 + (-0.0440 + 0.181i)T + (-4.44 - 2.29i)T^{2} \)
11 \( 1 + (5.75 + 0.273i)T + (10.9 + 1.04i)T^{2} \)
13 \( 1 + (0.543 + 0.471i)T + (1.85 + 12.8i)T^{2} \)
17 \( 1 + (-2.18 + 3.06i)T + (-5.56 - 16.0i)T^{2} \)
19 \( 1 + (0.00719 + 0.0100i)T + (-6.21 + 17.9i)T^{2} \)
29 \( 1 + (3.60 + 7.89i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (1.42 + 1.81i)T + (-7.30 + 30.1i)T^{2} \)
37 \( 1 + (-1.09 - 1.14i)T + (-1.76 + 36.9i)T^{2} \)
41 \( 1 + (1.29 + 4.41i)T + (-34.4 + 22.1i)T^{2} \)
43 \( 1 + (5.05 + 0.726i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + (4.15 + 2.39i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.92 + 3.43i)T + (41.6 + 32.7i)T^{2} \)
59 \( 1 + (-0.0414 + 0.214i)T + (-54.7 - 21.9i)T^{2} \)
61 \( 1 + (-8.68 - 3.47i)T + (44.1 + 42.0i)T^{2} \)
67 \( 1 + (-2.85 + 5.53i)T + (-38.8 - 54.5i)T^{2} \)
71 \( 1 + (-0.637 + 0.409i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-1.16 - 12.2i)T + (-71.6 + 13.8i)T^{2} \)
79 \( 1 + (0.308 - 0.106i)T + (62.0 - 48.8i)T^{2} \)
83 \( 1 + (-14.8 - 4.34i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (-10.2 - 8.07i)T + (20.9 + 86.4i)T^{2} \)
97 \( 1 + (-11.4 + 3.36i)T + (81.6 - 52.4i)T^{2} \)
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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

−10.26013870553414041804818639225, −9.548864609228933062230782572899, −8.059534639490809244859300818089, −7.63964288688841873405102927982, −6.87749558225762034275675793560, −5.61605880011324438322500910806, −4.88283291022048501976196120139, −3.49436565537112814684456368799, −2.03319841098392088204553642663, −0.59046879645628920716725796389, 2.01525771365751778845288678973, 3.25597397039988849692242827695, 4.83912992765969717211948629317, 5.07123128602357333708975023023, 6.21741126337838268485372887467, 7.57665777386016294578221103566, 8.273319549776759517620609080424, 9.218832842001517460819578053021, 10.40330238379323098955852384984, 10.56603285049300430793823128782

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