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Properties

Label 2-644-7.4-c3-0-17
Degree $2$
Conductor $644$
Sign $0.989 - 0.147i$
Analytic cond. $37.9972$
Root an. cond. $6.16418$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.29 − 3.98i)3-s + (4.80 + 8.32i)5-s + (−17.6 − 5.68i)7-s + (2.93 + 5.08i)9-s + (−3.25 + 5.63i)11-s − 52.1·13-s + 44.1·15-s + (55.0 − 95.2i)17-s + (38.6 + 66.9i)19-s + (−63.1 + 57.1i)21-s + (11.5 + 19.9i)23-s + (16.2 − 28.2i)25-s + 151.·27-s + 238.·29-s + (−90.6 + 157. i)31-s + ⋯
L(s)  = 1  + (0.442 − 0.766i)3-s + (0.429 + 0.744i)5-s + (−0.951 − 0.306i)7-s + (0.108 + 0.188i)9-s + (−0.0892 + 0.154i)11-s − 1.11·13-s + 0.760·15-s + (0.784 − 1.35i)17-s + (0.466 + 0.807i)19-s + (−0.656 + 0.593i)21-s + (0.104 + 0.180i)23-s + (0.130 − 0.225i)25-s + 1.07·27-s + 1.52·29-s + (−0.525 + 0.909i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.147i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.989 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.989 - 0.147i$
Analytic conductor: \(37.9972\)
Root analytic conductor: \(6.16418\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :3/2),\ 0.989 - 0.147i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.232553343\)
\(L(\frac12)\) \(\approx\) \(2.232553343\)
\(L(\frac{5}{2})\) 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 \( 1 + (17.6 + 5.68i)T \)
23 \( 1 + (-11.5 - 19.9i)T \)
good3 \( 1 + (-2.29 + 3.98i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-4.80 - 8.32i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (3.25 - 5.63i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 52.1T + 2.19e3T^{2} \)
17 \( 1 + (-55.0 + 95.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-38.6 - 66.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
29 \( 1 - 238.T + 2.43e4T^{2} \)
31 \( 1 + (90.6 - 157. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-129. - 224. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 173.T + 6.89e4T^{2} \)
43 \( 1 - 379.T + 7.95e4T^{2} \)
47 \( 1 + (19.5 + 33.9i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (336. - 582. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-302. + 523. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-253. - 438. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-61.3 + 106. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 219.T + 3.57e5T^{2} \)
73 \( 1 + (536. - 928. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-56.7 - 98.3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 167.T + 5.71e5T^{2} \)
89 \( 1 + (-618. - 1.07e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 343.T + 9.12e5T^{2} \)
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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

−9.947949273343719351366546701283, −9.640885164458413099508452440347, −8.250141126853895105604381726181, −7.23450268860011256294971262102, −6.99732662210352088473157133221, −5.84625698111231029520161454517, −4.65424254695416432250805601835, −3.03555467738632148772171677449, −2.56070809047935435555271019046, −1.01232908052784125921381176023, 0.74802556791669895947175260428, 2.46114339065777128735611122190, 3.50161783436880906351863818697, 4.53024819788033204335661596553, 5.50893341597005233185134699013, 6.43887390612240087512682197211, 7.60914514200676623486323031009, 8.742276185193117040565299515352, 9.378307510455634428588712607037, 9.873738928718181708042403225196

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