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Properties

Label 2-644-7.2-c3-0-0
Degree $2$
Conductor $644$
Sign $-0.874 + 0.485i$
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  + (1.08 + 1.87i)3-s + (−5.65 + 9.80i)5-s + (10.5 − 15.2i)7-s + (11.1 − 19.3i)9-s + (26.1 + 45.2i)11-s − 76.3·13-s − 24.5·15-s + (−59.2 − 102. i)17-s + (−52.8 + 91.4i)19-s + (40.0 + 3.22i)21-s + (11.5 − 19.9i)23-s + (−1.54 − 2.66i)25-s + 106.·27-s − 187.·29-s + (22.8 + 39.5i)31-s + ⋯
L(s)  = 1  + (0.208 + 0.361i)3-s + (−0.506 + 0.876i)5-s + (0.567 − 0.823i)7-s + (0.413 − 0.715i)9-s + (0.715 + 1.23i)11-s − 1.62·13-s − 0.422·15-s + (−0.844 − 1.46i)17-s + (−0.637 + 1.10i)19-s + (0.415 + 0.0334i)21-s + (0.104 − 0.180i)23-s + (−0.0123 − 0.0213i)25-s + 0.761·27-s − 1.19·29-s + (0.132 + 0.229i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 + 0.485i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $-0.874 + 0.485i$
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: \((2,\ 644,\ (\ :3/2),\ -0.874 + 0.485i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.01222409107\)
\(L(\frac12)\) \(\approx\) \(0.01222409107\)
\(L(\frac{5}{2})\) 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 \( 1 + (-10.5 + 15.2i)T \)
23 \( 1 + (-11.5 + 19.9i)T \)
good3 \( 1 + (-1.08 - 1.87i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (5.65 - 9.80i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-26.1 - 45.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 76.3T + 2.19e3T^{2} \)
17 \( 1 + (59.2 + 102. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (52.8 - 91.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
29 \( 1 + 187.T + 2.43e4T^{2} \)
31 \( 1 + (-22.8 - 39.5i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-15.5 + 26.8i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 160.T + 6.89e4T^{2} \)
43 \( 1 + 9.23T + 7.95e4T^{2} \)
47 \( 1 + (306. - 531. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (333. + 577. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (262. + 455. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (335. - 580. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (159. + 276. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 445.T + 3.57e5T^{2} \)
73 \( 1 + (465. + 805. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-137. + 237. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 542.T + 5.71e5T^{2} \)
89 \( 1 + (-325. + 563. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 844.T + 9.12e5T^{2} \)
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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

−10.58738801659704598045120449919, −9.749554122568889216375571258890, −9.258797057293968889276159042567, −7.69645249740152501999334158655, −7.22839667687267245062174541655, −6.56261953873116237210038798953, −4.74990573450567384120166271324, −4.27623829101200046829065032087, −3.13736318459069315775867667116, −1.80222734034926398373215941099, 0.00322310154359471646617788481, 1.55686447803643572415283125654, 2.58325486323470255584706631426, 4.21497193543292711930506260593, 4.91228375242254070193448600686, 5.97729571430709845238918129560, 7.13377106742892613545615135834, 8.130682758329998951347690595441, 8.630628550412521032964807464220, 9.365739459553349911127965442003

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