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Properties

Label 2-644-7.4-c1-0-0
Degree $2$
Conductor $644$
Sign $-0.908 + 0.417i$
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  + (−1.03 + 1.80i)3-s + (0.832 + 1.44i)5-s + (−2.64 − 0.0784i)7-s + (−0.661 − 1.14i)9-s + (−0.494 + 0.856i)11-s − 2.84·13-s − 3.46·15-s + (−1.02 + 1.76i)17-s + (−1.57 − 2.72i)19-s + (2.89 − 4.68i)21-s + (0.5 + 0.866i)23-s + (1.11 − 1.92i)25-s − 3.48·27-s + 3.06·29-s + (−3.31 + 5.74i)31-s + ⋯
L(s)  = 1  + (−0.600 + 1.03i)3-s + (0.372 + 0.645i)5-s + (−0.999 − 0.0296i)7-s + (−0.220 − 0.381i)9-s + (−0.149 + 0.258i)11-s − 0.787·13-s − 0.894·15-s + (−0.247 + 0.429i)17-s + (−0.360 − 0.624i)19-s + (0.630 − 1.02i)21-s + (0.104 + 0.180i)23-s + (0.222 − 0.385i)25-s − 0.671·27-s + 0.569·29-s + (−0.595 + 1.03i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $-0.908 + 0.417i$
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: \((2,\ 644,\ (\ :1/2),\ -0.908 + 0.417i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0971415 - 0.444247i\)
\(L(\frac12)\) \(\approx\) \(0.0971415 - 0.444247i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 + 0.0784i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (1.03 - 1.80i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.832 - 1.44i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.494 - 0.856i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.84T + 13T^{2} \)
17 \( 1 + (1.02 - 1.76i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.57 + 2.72i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 - 3.06T + 29T^{2} \)
31 \( 1 + (3.31 - 5.74i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.00 + 6.93i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.47T + 41T^{2} \)
43 \( 1 + 4.79T + 43T^{2} \)
47 \( 1 + (5.26 + 9.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.05 - 1.82i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.30 - 9.18i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.886 - 1.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.30 - 4.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.86T + 71T^{2} \)
73 \( 1 + (5.12 - 8.87i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.641 + 1.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.51T + 83T^{2} \)
89 \( 1 + (-4.35 - 7.54i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.2T + 97T^{2} \)
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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

−10.60348363931793620761492416100, −10.38880459620785105782599857155, −9.597864553459067846435392286095, −8.756041468152894779633772763453, −7.25497403141186581393911499995, −6.56842061048703848850045337884, −5.55725785718206917002429103287, −4.66057954528840219384735031196, −3.57322499326589741365377459174, −2.41402823795434778006181367836, 0.25156630737583062156299459233, 1.72424699580955389120958327400, 3.13909944558040536179657057334, 4.68688514632302902027277503695, 5.74082388752795398320739882696, 6.45948737876033328333831360261, 7.23121762038269408071419573593, 8.226436049265186992345026558770, 9.336284704685724325329725052143, 9.920524040120997870303889043728

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