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Properties

Label 2-644-7.2-c3-0-24
Degree $2$
Conductor $644$
Sign $0.0222 - 0.999i$
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  + (1.35 + 2.35i)3-s + (−4.93 + 8.55i)5-s + (7.85 + 16.7i)7-s + (9.81 − 17.0i)9-s + (21.8 + 37.8i)11-s + 79.7·13-s − 26.7·15-s + (−1.13 − 1.95i)17-s + (45.5 − 78.9i)19-s + (−28.7 + 41.2i)21-s + (11.5 − 19.9i)23-s + (13.7 + 23.7i)25-s + 126.·27-s + 27.2·29-s + (−65.3 − 113. i)31-s + ⋯
L(s)  = 1  + (0.261 + 0.452i)3-s + (−0.441 + 0.764i)5-s + (0.424 + 0.905i)7-s + (0.363 − 0.629i)9-s + (0.599 + 1.03i)11-s + 1.70·13-s − 0.461·15-s + (−0.0161 − 0.0279i)17-s + (0.550 − 0.953i)19-s + (−0.298 + 0.428i)21-s + (0.104 − 0.180i)23-s + (0.109 + 0.190i)25-s + 0.902·27-s + 0.174·29-s + (−0.378 − 0.656i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0222 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0222 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.0222 - 0.999i$
Analytic conductor: \(37.9972\)
Root analytic conductor: \(6.16418\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :3/2),\ 0.0222 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.561949443\)
\(L(\frac12)\) \(\approx\) \(2.561949443\)
\(L(\frac{5}{2})\) 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 \( 1 + (-7.85 - 16.7i)T \)
23 \( 1 + (-11.5 + 19.9i)T \)
good3 \( 1 + (-1.35 - 2.35i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (4.93 - 8.55i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-21.8 - 37.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 79.7T + 2.19e3T^{2} \)
17 \( 1 + (1.13 + 1.95i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-45.5 + 78.9i)T + (-3.42e3 - 5.94e3i)T^{2} \)
29 \( 1 - 27.2T + 2.43e4T^{2} \)
31 \( 1 + (65.3 + 113. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (25.4 - 44.0i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 364.T + 6.89e4T^{2} \)
43 \( 1 + 227.T + 7.95e4T^{2} \)
47 \( 1 + (110. - 190. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-237. - 411. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-139. - 241. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (150. - 260. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (119. + 206. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 233.T + 3.57e5T^{2} \)
73 \( 1 + (463. + 802. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-302. + 523. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 375.T + 5.71e5T^{2} \)
89 \( 1 + (516. - 893. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 541.T + 9.12e5T^{2} \)
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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

−10.43057688328200922451379116643, −9.256460998007332803137215067936, −8.953305559337341736914159673667, −7.73177977624170560182388221546, −6.81016057704514010593969772629, −5.99155401396568820566498193249, −4.64962949450241228821034612087, −3.76245340861207905619631493718, −2.78489960426906469767400731232, −1.31073788414686725801725702957, 0.871192142311804640958027283396, 1.53190897219536844035257343948, 3.45711678247175689326226789114, 4.18020056548551871131442011863, 5.35105656461867068969870727280, 6.45035339472652144511455183108, 7.46424037984200993858227199691, 8.371477956469107621464756127960, 8.623435230245845892334305747277, 10.03240095902412709584873155458

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