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Properties

Label 2-770-385.54-c1-0-30
Degree $2$
Conductor $770$
Sign $0.999 + 0.000420i$
Analytic cond. $6.14848$
Root an. cond. $2.47961$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (1.44 + 2.49i)3-s + (−0.499 − 0.866i)4-s + (−1.57 − 1.58i)5-s − 2.88·6-s + (−0.391 − 2.61i)7-s + 0.999·8-s + (−2.64 + 4.58i)9-s + (2.16 − 0.568i)10-s + (0.914 − 3.18i)11-s + (1.44 − 2.49i)12-s − 6.76i·13-s + (2.46 + 0.969i)14-s + (1.69 − 6.21i)15-s + (−0.5 + 0.866i)16-s + (1.33 − 0.772i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.831 + 1.44i)3-s + (−0.249 − 0.433i)4-s + (−0.703 − 0.710i)5-s − 1.17·6-s + (−0.147 − 0.989i)7-s + 0.353·8-s + (−0.882 + 1.52i)9-s + (0.683 − 0.179i)10-s + (0.275 − 0.961i)11-s + (0.415 − 0.720i)12-s − 1.87i·13-s + (0.657 + 0.259i)14-s + (0.437 − 1.60i)15-s + (−0.125 + 0.216i)16-s + (0.324 − 0.187i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.000420i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.000420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.999 + 0.000420i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.999 + 0.000420i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25662 - 0.000264117i\)
\(L(\frac12)\) \(\approx\) \(1.25662 - 0.000264117i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (1.57 + 1.58i)T \)
7 \( 1 + (0.391 + 2.61i)T \)
11 \( 1 + (-0.914 + 3.18i)T \)
good3 \( 1 + (-1.44 - 2.49i)T + (-1.5 + 2.59i)T^{2} \)
13 \( 1 + 6.76iT - 13T^{2} \)
17 \( 1 + (-1.33 + 0.772i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.445 - 0.771i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.17 - 3.56i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.79iT - 29T^{2} \)
31 \( 1 + (-0.849 + 0.490i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.14 + 4.70i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.03T + 41T^{2} \)
43 \( 1 - 0.302T + 43T^{2} \)
47 \( 1 + (-5.24 + 9.08i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.89 + 2.82i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.61 + 2.08i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.226 - 0.393i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.39 - 4.84i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.72T + 71T^{2} \)
73 \( 1 + (-2.80 + 1.61i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.09 - 4.09i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.01iT - 83T^{2} \)
89 \( 1 + (-6.83 - 3.94i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 19.1T + 97T^{2} \)
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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

−10.32586128218112980429086691430, −9.296275649947990639207822458311, −8.604090502993027057811926040843, −8.018055938393303329953499774877, −7.20044115378542341173457171761, −5.55525970491311269340670571008, −4.93890900236435447131019608395, −3.72960208828036957034544505377, −3.31560101880407677192155034399, −0.68046908781728473225295443649, 1.60379262251853188393764298274, 2.44052069509704195617066162040, 3.33644936554168541293920488206, 4.63606902381350267641487084841, 6.54268684478942720648323234517, 6.91389291479148927053168035665, 7.73973377692692927765245904034, 8.853796478703037685702383721813, 9.007738209504576558879757671640, 10.31521101435484903864411648420

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