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Properties

Label 2-644-7.4-c3-0-7
Degree $2$
Conductor $644$
Sign $0.506 - 0.862i$
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  + (−2.07 + 3.58i)3-s + (−7.90 − 13.6i)5-s + (−15.5 + 10.1i)7-s + (4.92 + 8.53i)9-s + (5.27 − 9.13i)11-s − 38.8·13-s + 65.4·15-s + (54.9 − 95.2i)17-s + (−58.8 − 101. i)19-s + (−4.25 − 76.5i)21-s + (11.5 + 19.9i)23-s + (−62.4 + 108. i)25-s − 152.·27-s − 0.117·29-s + (−135. + 235. i)31-s + ⋯
L(s)  = 1  + (−0.398 + 0.690i)3-s + (−0.706 − 1.22i)5-s + (−0.836 + 0.547i)7-s + (0.182 + 0.316i)9-s + (0.144 − 0.250i)11-s − 0.828·13-s + 1.12·15-s + (0.784 − 1.35i)17-s + (−0.710 − 1.23i)19-s + (−0.0442 − 0.795i)21-s + (0.104 + 0.180i)23-s + (−0.499 + 0.865i)25-s − 1.08·27-s − 0.000752·29-s + (−0.787 + 1.36i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 - 0.862i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.506 - 0.862i$
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: \((2,\ 644,\ (\ :3/2),\ 0.506 - 0.862i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8407458385\)
\(L(\frac12)\) \(\approx\) \(0.8407458385\)
\(L(\frac{5}{2})\) 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 \( 1 + (15.5 - 10.1i)T \)
23 \( 1 + (-11.5 - 19.9i)T \)
good3 \( 1 + (2.07 - 3.58i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (7.90 + 13.6i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-5.27 + 9.13i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 38.8T + 2.19e3T^{2} \)
17 \( 1 + (-54.9 + 95.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (58.8 + 101. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
29 \( 1 + 0.117T + 2.43e4T^{2} \)
31 \( 1 + (135. - 235. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-85.4 - 147. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 76.8T + 6.89e4T^{2} \)
43 \( 1 - 213.T + 7.95e4T^{2} \)
47 \( 1 + (-163. - 282. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-226. + 392. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-35.2 + 61.1i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-37.2 - 64.5i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (490. - 850. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 1.09e3T + 3.57e5T^{2} \)
73 \( 1 + (323. - 560. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-690. - 1.19e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 806.T + 5.71e5T^{2} \)
89 \( 1 + (-143. - 249. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 908.T + 9.12e5T^{2} \)
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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

−10.18224012444296942401225905873, −9.364875194271054660416749425182, −8.838637686814928094250354715321, −7.72077426800300843885059839295, −6.78364200963847959629339853352, −5.29897278696310984002382901118, −4.97836052822397861849914034310, −3.91639877953920855013647440123, −2.63594921636974237598652531449, −0.73010249026269659986949871888, 0.39186539111748676679939191855, 2.02582891135276173503636175546, 3.48823820754130928127734194757, 4.04528629721611163223984167733, 5.94985171505084808616517594312, 6.45050857166420623527824044823, 7.44871442736464359416659719199, 7.72628503882629976813581189397, 9.314945285371761003885014432385, 10.28752075020868249928173283294

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