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Properties

Label 2-644-7.4-c3-0-15
Degree $2$
Conductor $644$
Sign $0.968 - 0.247i$
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  + (−4.78 + 8.29i)3-s + (1.66 + 2.89i)5-s + (−18.1 − 3.85i)7-s + (−32.3 − 56.0i)9-s + (−11.8 + 20.5i)11-s − 40.9·13-s − 31.9·15-s + (−12.8 + 22.3i)17-s + (3.14 + 5.44i)19-s + (118. − 131. i)21-s + (11.5 + 19.9i)23-s + (56.9 − 98.5i)25-s + 360.·27-s − 194.·29-s + (118. − 205. i)31-s + ⋯
L(s)  = 1  + (−0.921 + 1.59i)3-s + (0.149 + 0.258i)5-s + (−0.978 − 0.208i)7-s + (−1.19 − 2.07i)9-s + (−0.325 + 0.563i)11-s − 0.873·13-s − 0.550·15-s + (−0.183 + 0.318i)17-s + (0.0379 + 0.0657i)19-s + (1.23 − 1.36i)21-s + (0.104 + 0.180i)23-s + (0.455 − 0.788i)25-s + 2.57·27-s − 1.24·29-s + (0.689 − 1.19i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.968 - 0.247i$
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: \((2,\ 644,\ (\ :3/2),\ 0.968 - 0.247i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5240809377\)
\(L(\frac12)\) \(\approx\) \(0.5240809377\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.1 + 3.85i)T \)
23 \( 1 + (-11.5 - 19.9i)T \)
good3 \( 1 + (4.78 - 8.29i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-1.66 - 2.89i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (11.8 - 20.5i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 40.9T + 2.19e3T^{2} \)
17 \( 1 + (12.8 - 22.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-3.14 - 5.44i)T + (-3.42e3 + 5.94e3i)T^{2} \)
29 \( 1 + 194.T + 2.43e4T^{2} \)
31 \( 1 + (-118. + 205. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (25.8 + 44.6i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 346.T + 6.89e4T^{2} \)
43 \( 1 - 68.3T + 7.95e4T^{2} \)
47 \( 1 + (-214. - 371. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (29.3 - 50.8i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-300. + 520. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-387. - 670. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (509. - 883. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 534.T + 3.57e5T^{2} \)
73 \( 1 + (-227. + 393. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-3.80 - 6.59i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.16e3T + 5.71e5T^{2} \)
89 \( 1 + (533. + 923. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 511.T + 9.12e5T^{2} \)
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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

−10.06919932907055901612600375668, −9.762219164986154702158406506034, −8.866653635546576722350245070294, −7.36888344861739081291655284105, −6.36094506858497408194613874592, −5.58364289542344381529347187152, −4.60511095419641604155083948118, −3.81872862450891282705181800226, −2.66616311123042885735379693454, −0.26922987402237115271365278674, 0.67386866435607586355580013039, 2.00656864087233713952849539301, 3.12613750186292547641160165371, 5.07488730361251018020563206640, 5.68129359888982625741485759658, 6.71796501866889532629394770880, 7.14913115154266971856949945978, 8.211201727886438632015231790778, 9.165100146688704540658498897749, 10.30024990894763738830998276848

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