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Properties

Label 2-77-7.5-c4-0-10
Degree $2$
Conductor $77$
Sign $0.824 - 0.565i$
Analytic cond. $7.95948$
Root an. cond. $2.82125$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.21 − 2.09i)2-s + (0.957 − 0.552i)3-s + (5.06 + 8.77i)4-s + (15.8 + 9.13i)5-s − 2.67i·6-s + (−36.5 + 32.6i)7-s + 63.3·8-s + (−39.8 + 69.0i)9-s + (38.3 − 22.1i)10-s + (18.2 + 31.5i)11-s + (9.69 + 5.59i)12-s + 40.0i·13-s + (24.1 + 116. i)14-s + 20.1·15-s + (−4.37 + 7.57i)16-s + (357. − 206. i)17-s + ⋯
L(s)  = 1  + (0.302 − 0.524i)2-s + (0.106 − 0.0614i)3-s + (0.316 + 0.548i)4-s + (0.632 + 0.365i)5-s − 0.0743i·6-s + (−0.746 + 0.665i)7-s + 0.989·8-s + (−0.492 + 0.852i)9-s + (0.383 − 0.221i)10-s + (0.150 + 0.261i)11-s + (0.0673 + 0.0388i)12-s + 0.236i·13-s + (0.123 + 0.592i)14-s + 0.0897·15-s + (−0.0170 + 0.0296i)16-s + (1.23 − 0.713i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.824 - 0.565i$
Analytic conductor: \(7.95948\)
Root analytic conductor: \(2.82125\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :2),\ 0.824 - 0.565i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.13043 + 0.660782i\)
\(L(\frac12)\) \(\approx\) \(2.13043 + 0.660782i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (36.5 - 32.6i)T \)
11 \( 1 + (-18.2 - 31.5i)T \)
good2 \( 1 + (-1.21 + 2.09i)T + (-8 - 13.8i)T^{2} \)
3 \( 1 + (-0.957 + 0.552i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-15.8 - 9.13i)T + (312.5 + 541. i)T^{2} \)
13 \( 1 - 40.0iT - 2.85e4T^{2} \)
17 \( 1 + (-357. + 206. i)T + (4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (-199. - 115. i)T + (6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (-305. + 529. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 - 3.36T + 7.07e5T^{2} \)
31 \( 1 + (870. - 502. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + (-420. + 727. i)T + (-9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 - 349. iT - 2.82e6T^{2} \)
43 \( 1 + 1.10e3T + 3.41e6T^{2} \)
47 \( 1 + (809. + 467. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (1.72e3 + 2.97e3i)T + (-3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-880. + 508. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-1.64e3 - 947. i)T + (6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (968. + 1.67e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 739.T + 2.54e7T^{2} \)
73 \( 1 + (-8.70e3 + 5.02e3i)T + (1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (-1.32e3 + 2.29e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 9.44e3iT - 4.74e7T^{2} \)
89 \( 1 + (-9.09e3 - 5.24e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 - 1.51e4iT - 8.85e7T^{2} \)
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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

−13.70316031501519470755675044014, −12.67216827696292556165442769749, −11.81002899097838103833613496219, −10.62861576997098343572480820240, −9.522456288292250318413790555465, −8.048658296283532117012846141937, −6.74003801770641571220370938277, −5.25558591950747552054322065025, −3.25213382389421829805272697763, −2.19370591543684625374154466339, 1.12375914351315605066548926839, 3.48295300678390649347531070706, 5.42451297995698661339089826699, 6.27475512647309745391094415272, 7.54381297119243518039414536389, 9.309730441967015159596932523632, 10.08488175210707584036161201891, 11.39316020378624778086864139294, 12.87471099580693916448966530703, 13.77666677885471660378213150030

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