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Properties

Label 2-644-92.91-c1-0-55
Degree $2$
Conductor $644$
Sign $-0.796 - 0.604i$
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  + (−0.340 − 1.37i)2-s + 0.155i·3-s + (−1.76 + 0.934i)4-s + 1.40i·5-s + (0.212 − 0.0527i)6-s − 7-s + (1.88 + 2.10i)8-s + 2.97·9-s + (1.92 − 0.478i)10-s − 3.42·11-s + (−0.144 − 0.274i)12-s − 5.81·13-s + (0.340 + 1.37i)14-s − 0.217·15-s + (2.25 − 3.30i)16-s − 4.21i·17-s + ⋯
L(s)  = 1  + (−0.240 − 0.970i)2-s + 0.0895i·3-s + (−0.884 + 0.467i)4-s + 0.628i·5-s + (0.0868 − 0.0215i)6-s − 0.377·7-s + (0.666 + 0.745i)8-s + 0.991·9-s + (0.609 − 0.151i)10-s − 1.03·11-s + (−0.0418 − 0.0791i)12-s − 1.61·13-s + (0.0909 + 0.366i)14-s − 0.0562·15-s + (0.563 − 0.826i)16-s − 1.02i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $-0.796 - 0.604i$
Analytic conductor: \(5.14236\)
Root analytic conductor: \(2.26767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :1/2),\ -0.796 - 0.604i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0131956 + 0.0391844i\)
\(L(\frac12)\) \(\approx\) \(0.0131956 + 0.0391844i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.340 + 1.37i)T \)
7 \( 1 + T \)
23 \( 1 + (2.02 + 4.34i)T \)
good3 \( 1 - 0.155iT - 3T^{2} \)
5 \( 1 - 1.40iT - 5T^{2} \)
11 \( 1 + 3.42T + 11T^{2} \)
13 \( 1 + 5.81T + 13T^{2} \)
17 \( 1 + 4.21iT - 17T^{2} \)
19 \( 1 + 7.13T + 19T^{2} \)
29 \( 1 + 5.12T + 29T^{2} \)
31 \( 1 + 4.81iT - 31T^{2} \)
37 \( 1 - 12.1iT - 37T^{2} \)
41 \( 1 + 2.16T + 41T^{2} \)
43 \( 1 + 5.69T + 43T^{2} \)
47 \( 1 - 1.80iT - 47T^{2} \)
53 \( 1 - 8.27iT - 53T^{2} \)
59 \( 1 + 5.17iT - 59T^{2} \)
61 \( 1 + 11.4iT - 61T^{2} \)
67 \( 1 - 8.18T + 67T^{2} \)
71 \( 1 - 11.6iT - 71T^{2} \)
73 \( 1 + 0.100T + 73T^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 + 2.68T + 83T^{2} \)
89 \( 1 - 3.82iT - 89T^{2} \)
97 \( 1 + 8.45iT - 97T^{2} \)
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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

−10.03889504797108367891028662508, −9.658130012509058653626906917411, −8.389500166500396608597990636765, −7.48132394964589396674342116152, −6.67157581561840216850195142494, −5.04481406464422261491889223948, −4.34284720200555722636098309205, −2.94495775836204837775399940770, −2.19502676248507880278711147671, −0.02280935296490578503226976071, 1.93975230481840024134872560888, 3.93794191259042032955746598078, 4.85907051602808999474581364721, 5.66562508934312071380038454404, 6.84737338641346143829800389614, 7.50563659076948026009974726884, 8.366843594227824513874862789971, 9.242727899493904684848991874076, 10.11588380006109878692353568898, 10.61179498096028346305314377841

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