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Properties

Label 2-770-385.164-c1-0-0
Degree $2$
Conductor $770$
Sign $-0.217 + 0.976i$
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 + (−0.617 + 1.06i)3-s + (−0.499 + 0.866i)4-s + (−1.46 + 1.68i)5-s − 1.23·6-s + (−1.62 − 2.09i)7-s − 0.999·8-s + (0.737 + 1.27i)9-s + (−2.19 − 0.429i)10-s + (−2.28 − 2.40i)11-s + (−0.617 − 1.06i)12-s + 4.24i·13-s + (1.00 − 2.44i)14-s + (−0.895 − 2.61i)15-s + (−0.5 − 0.866i)16-s + (2.39 + 1.38i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.356 + 0.617i)3-s + (−0.249 + 0.433i)4-s + (−0.656 + 0.753i)5-s − 0.504·6-s + (−0.612 − 0.790i)7-s − 0.353·8-s + (0.245 + 0.425i)9-s + (−0.693 − 0.135i)10-s + (−0.688 − 0.725i)11-s + (−0.178 − 0.308i)12-s + 1.17i·13-s + (0.267 − 0.654i)14-s + (−0.231 − 0.674i)15-s + (−0.125 − 0.216i)16-s + (0.581 + 0.335i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.217 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.217 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.217 + 0.976i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.217 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.132796 - 0.165676i\)
\(L(\frac12)\) \(\approx\) \(0.132796 - 0.165676i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.46 - 1.68i)T \)
7 \( 1 + (1.62 + 2.09i)T \)
11 \( 1 + (2.28 + 2.40i)T \)
good3 \( 1 + (0.617 - 1.06i)T + (-1.5 - 2.59i)T^{2} \)
13 \( 1 - 4.24iT - 13T^{2} \)
17 \( 1 + (-2.39 - 1.38i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.25 + 5.63i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.95 - 2.28i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.47iT - 29T^{2} \)
31 \( 1 + (2.20 + 1.27i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.58 + 0.917i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.92T + 41T^{2} \)
43 \( 1 + 0.711T + 43T^{2} \)
47 \( 1 + (-4.67 - 8.09i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.73 - 1.00i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.99 + 2.88i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.04 - 7.01i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.1 + 7.03i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.71T + 71T^{2} \)
73 \( 1 + (3.07 + 1.77i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.03 - 4.64i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.3iT - 83T^{2} \)
89 \( 1 + (15.9 - 9.23i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 15.4T + 97T^{2} \)
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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

−10.95810709773467780172887874553, −10.17252077086363342080391200589, −9.322912493326074643898630538014, −8.043065947252892991500864023860, −7.43274970109058735407026482197, −6.54525853218532281981594168053, −5.72792337389118504253680626197, −4.34286623174955646731222378578, −4.00581725113241976439328669236, −2.68573875976082523198579706171, 0.098000949442331038981611190255, 1.61725657679577966865769902004, 3.04032148863309260970464863127, 4.06861302136778268317433960068, 5.31073070208273084792279686847, 5.87663677764122667925680352195, 7.10282569547843774433045155739, 8.025420934493201494794845846946, 8.872254618169820694655655931143, 9.913898207593213293675395808543

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