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Properties

Label 2-644-7.4-c1-0-9
Degree $2$
Conductor $644$
Sign $0.479 + 0.877i$
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  + (1.09 − 1.89i)3-s + (0.0359 + 0.0623i)5-s + (0.105 + 2.64i)7-s + (−0.895 − 1.55i)9-s + (2.19 − 3.79i)11-s + 1.85·13-s + 0.157·15-s + (2.43 − 4.21i)17-s + (−0.857 − 1.48i)19-s + (5.12 + 2.69i)21-s + (0.5 + 0.866i)23-s + (2.49 − 4.32i)25-s + 2.64·27-s − 2.32·29-s + (−1.89 + 3.27i)31-s + ⋯
L(s)  = 1  + (0.631 − 1.09i)3-s + (0.0160 + 0.0278i)5-s + (0.0397 + 0.999i)7-s + (−0.298 − 0.517i)9-s + (0.660 − 1.14i)11-s + 0.513·13-s + 0.0406·15-s + (0.590 − 1.02i)17-s + (−0.196 − 0.340i)19-s + (1.11 + 0.587i)21-s + (0.104 + 0.180i)23-s + (0.499 − 0.865i)25-s + 0.509·27-s − 0.431·29-s + (−0.339 + 0.588i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.479 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.479 + 0.877i$
Analytic conductor: \(5.14236\)
Root analytic conductor: \(2.26767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :1/2),\ 0.479 + 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67777 - 0.995210i\)
\(L(\frac12)\) \(\approx\) \(1.67777 - 0.995210i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.105 - 2.64i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (-1.09 + 1.89i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.0359 - 0.0623i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.19 + 3.79i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.85T + 13T^{2} \)
17 \( 1 + (-2.43 + 4.21i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.857 + 1.48i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 2.32T + 29T^{2} \)
31 \( 1 + (1.89 - 3.27i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.57 - 4.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.793T + 41T^{2} \)
43 \( 1 + 8.05T + 43T^{2} \)
47 \( 1 + (2.52 + 4.37i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.179 + 0.310i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.42 - 2.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.95 - 8.57i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.18 - 10.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.98T + 71T^{2} \)
73 \( 1 + (-3.07 + 5.32i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.64 + 6.31i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 + (-8.92 - 15.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.62T + 97T^{2} \)
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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

−10.43967386552385623355449072945, −9.153797223732930443644950128671, −8.652292036773358713116145989926, −7.910927150707139982889824987331, −6.84747975972973510223446528525, −6.12347273498251553641648816770, −5.04779289644721497611946448653, −3.38614940974970863061280065074, −2.51441671176638283402080675342, −1.17642140625725626389615806198, 1.62135220856097930246435528446, 3.43589076784993242844886680854, 4.01463630605461649151326091197, 4.86930776408355520785739307707, 6.26782137868453298667956416824, 7.29371957775163180182845357149, 8.199437103687098155278611804643, 9.209324376305920581001709000198, 9.815472645611119883335153742481, 10.51211888309332968657945649461

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