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Properties

Label 2-644-7.4-c1-0-7
Degree $2$
Conductor $644$
Sign $0.937 + 0.348i$
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  + (−1.73 + 2.99i)3-s + (−1.14 − 1.97i)5-s + (1.81 + 1.92i)7-s + (−4.48 − 7.76i)9-s + (2.26 − 3.91i)11-s − 2.16·13-s + 7.90·15-s + (0.211 − 0.365i)17-s + (−3.39 − 5.87i)19-s + (−8.91 + 2.09i)21-s + (0.5 + 0.866i)23-s + (−0.111 + 0.193i)25-s + 20.6·27-s + 8.14·29-s + (−1.48 + 2.57i)31-s + ⋯
L(s)  = 1  + (−0.998 + 1.73i)3-s + (−0.511 − 0.885i)5-s + (0.684 + 0.728i)7-s + (−1.49 − 2.58i)9-s + (0.681 − 1.18i)11-s − 0.599·13-s + 2.04·15-s + (0.0511 − 0.0886i)17-s + (−0.777 − 1.34i)19-s + (−1.94 + 0.456i)21-s + (0.104 + 0.180i)23-s + (−0.0223 + 0.0387i)25-s + 3.97·27-s + 1.51·29-s + (−0.266 + 0.461i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 + 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.937 + 0.348i$
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: \((2,\ 644,\ (\ :1/2),\ 0.937 + 0.348i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.769063 - 0.138448i\)
\(L(\frac12)\) \(\approx\) \(0.769063 - 0.138448i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.81 - 1.92i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (1.73 - 2.99i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.14 + 1.97i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.26 + 3.91i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.16T + 13T^{2} \)
17 \( 1 + (-0.211 + 0.365i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.39 + 5.87i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 - 8.14T + 29T^{2} \)
31 \( 1 + (1.48 - 2.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.54 + 4.40i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.35T + 41T^{2} \)
43 \( 1 - 0.589T + 43T^{2} \)
47 \( 1 + (-4.48 - 7.76i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.53 + 4.39i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.16 + 7.20i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.14 + 5.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.0488 - 0.0846i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.55T + 71T^{2} \)
73 \( 1 + (-3.07 + 5.33i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.86 + 11.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.02T + 83T^{2} \)
89 \( 1 + (2.51 + 4.35i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.16T + 97T^{2} \)
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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

−10.75209480749750560177562158260, −9.583676896581017071898037264126, −8.814430985287894941415744062364, −8.478178938164486036018386686524, −6.61050662086898528866830843018, −5.64217447908469254022386963347, −4.84975343366094733532390814594, −4.34775384891328597587839291670, −3.10491947083453249103823995601, −0.53657697755006924555575607484, 1.31095121575825435364515459805, 2.38503643564065641916063359581, 4.19198996989194835154852842950, 5.31133806570581459970624682107, 6.56425374405531335553570154788, 6.97792275540053458233395273119, 7.66238965205697323361810283266, 8.380840130053162793975799668122, 10.26399182430108719140054471202, 10.68886842083557483091275703650

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