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Properties

Label 2-770-35.27-c1-0-9
Degree $2$
Conductor $770$
Sign $0.996 - 0.0831i$
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  + (0.707 + 0.707i)2-s + (−2.10 − 2.10i)3-s + 1.00i·4-s + (−2.14 + 0.625i)5-s − 2.97i·6-s + (−2.64 + 0.0831i)7-s + (−0.707 + 0.707i)8-s + 5.87i·9-s + (−1.96 − 1.07i)10-s + 11-s + (2.10 − 2.10i)12-s + (1.45 + 1.45i)13-s + (−1.92 − 1.81i)14-s + (5.83 + 3.20i)15-s − 1.00·16-s + (4.24 − 4.24i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−1.21 − 1.21i)3-s + 0.500i·4-s + (−0.960 + 0.279i)5-s − 1.21i·6-s + (−0.999 + 0.0314i)7-s + (−0.250 + 0.250i)8-s + 1.95i·9-s + (−0.619 − 0.340i)10-s + 0.301·11-s + (0.608 − 0.608i)12-s + (0.404 + 0.404i)13-s + (−0.515 − 0.484i)14-s + (1.50 + 0.827i)15-s − 0.250·16-s + (1.02 − 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.996 - 0.0831i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.996 - 0.0831i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.901241 + 0.0375149i\)
\(L(\frac12)\) \(\approx\) \(0.901241 + 0.0375149i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (2.14 - 0.625i)T \)
7 \( 1 + (2.64 - 0.0831i)T \)
11 \( 1 - T \)
good3 \( 1 + (2.10 + 2.10i)T + 3iT^{2} \)
13 \( 1 + (-1.45 - 1.45i)T + 13iT^{2} \)
17 \( 1 + (-4.24 + 4.24i)T - 17iT^{2} \)
19 \( 1 - 1.23T + 19T^{2} \)
23 \( 1 + (-0.927 + 0.927i)T - 23iT^{2} \)
29 \( 1 - 1.51iT - 29T^{2} \)
31 \( 1 + 0.752iT - 31T^{2} \)
37 \( 1 + (-1.72 - 1.72i)T + 37iT^{2} \)
41 \( 1 - 6.54iT - 41T^{2} \)
43 \( 1 + (-8.39 + 8.39i)T - 43iT^{2} \)
47 \( 1 + (-3.36 + 3.36i)T - 47iT^{2} \)
53 \( 1 + (2.89 - 2.89i)T - 53iT^{2} \)
59 \( 1 - 3.48T + 59T^{2} \)
61 \( 1 - 12.0iT - 61T^{2} \)
67 \( 1 + (7.59 + 7.59i)T + 67iT^{2} \)
71 \( 1 - 4.25T + 71T^{2} \)
73 \( 1 + (-7.19 - 7.19i)T + 73iT^{2} \)
79 \( 1 - 16.7iT - 79T^{2} \)
83 \( 1 + (-8.61 - 8.61i)T + 83iT^{2} \)
89 \( 1 - 9.94T + 89T^{2} \)
97 \( 1 + (4.96 - 4.96i)T - 97iT^{2} \)
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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

−10.70025577518318325385423447883, −9.429181331832186101173333439464, −8.196578678982073311144724893758, −7.25515136828905608236383193968, −6.91442183873166752325124221482, −6.06772226315206004216920198786, −5.25346431433251487488395997968, −4.00496711963075085437618323030, −2.81891236316273581204306760319, −0.799479683205122827651736285828, 0.74268562483669654252772193349, 3.34868020763999087098678616022, 3.82762495508106657451685260302, 4.75282468679023838611790975573, 5.73535681891669122988943943731, 6.32995920291977878789924073248, 7.65923507907704046014566690988, 9.006141329048999256164148993624, 9.730437122377649727212682404091, 10.56126533134003193621966782536

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