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Properties

Label 2-770-55.32-c1-0-3
Degree $2$
Conductor $770$
Sign $-0.149 - 0.988i$
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 + (−1.98 + 1.98i)3-s − 1.00i·4-s + (−1.96 − 1.07i)5-s + 2.80i·6-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s − 4.88i·9-s + (−2.14 + 0.629i)10-s + (3.30 − 0.320i)11-s + (1.98 + 1.98i)12-s + (3.13 + 3.13i)13-s − 1.00i·14-s + (6.02 − 1.76i)15-s − 1.00·16-s + (−3.61 + 3.61i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−1.14 + 1.14i)3-s − 0.500i·4-s + (−0.877 − 0.479i)5-s + 1.14i·6-s + (0.267 − 0.267i)7-s + (−0.250 − 0.250i)8-s − 1.62i·9-s + (−0.678 + 0.199i)10-s + (0.995 − 0.0966i)11-s + (0.573 + 0.573i)12-s + (0.870 + 0.870i)13-s − 0.267i·14-s + (1.55 − 0.456i)15-s − 0.250·16-s + (−0.876 + 0.876i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.149 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.149 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.149 - 0.988i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.149 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.485521 + 0.564666i\)
\(L(\frac12)\) \(\approx\) \(0.485521 + 0.564666i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.96 + 1.07i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + (-3.30 + 0.320i)T \)
good3 \( 1 + (1.98 - 1.98i)T - 3iT^{2} \)
13 \( 1 + (-3.13 - 3.13i)T + 13iT^{2} \)
17 \( 1 + (3.61 - 3.61i)T - 17iT^{2} \)
19 \( 1 + 4.73T + 19T^{2} \)
23 \( 1 + (5.14 - 5.14i)T - 23iT^{2} \)
29 \( 1 - 1.00T + 29T^{2} \)
31 \( 1 + 1.88T + 31T^{2} \)
37 \( 1 + (-5.35 - 5.35i)T + 37iT^{2} \)
41 \( 1 - 11.8iT - 41T^{2} \)
43 \( 1 + (3.64 + 3.64i)T + 43iT^{2} \)
47 \( 1 + (-0.168 - 0.168i)T + 47iT^{2} \)
53 \( 1 + (4.47 - 4.47i)T - 53iT^{2} \)
59 \( 1 - 5.12iT - 59T^{2} \)
61 \( 1 + 7.34iT - 61T^{2} \)
67 \( 1 + (-7.29 - 7.29i)T + 67iT^{2} \)
71 \( 1 + 1.09T + 71T^{2} \)
73 \( 1 + (-1.66 - 1.66i)T + 73iT^{2} \)
79 \( 1 + 6.58T + 79T^{2} \)
83 \( 1 + (-4.50 - 4.50i)T + 83iT^{2} \)
89 \( 1 - 10.5iT - 89T^{2} \)
97 \( 1 + (-7.77 - 7.77i)T + 97iT^{2} \)
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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

−10.89761700897071269825782051284, −9.921681340706820208605525292458, −9.078382727699229010706327338943, −8.212280352590299038477096551889, −6.59632070291546870232300938687, −6.03769320726825379979837736688, −4.79563058645394787650727265868, −4.12681121301264432827319960851, −3.77224128908686882951142157832, −1.43976200430326913961393421869, 0.39188665397407031052635579193, 2.22267283435715196856486571288, 3.81025370134428808319300299829, 4.78518517487048461393423121074, 6.01002753375985346617354639678, 6.47175134158746660360771322266, 7.22298436887697114758962590669, 8.061019128363729762962188739073, 8.840365713500490107312384823783, 10.56504906583291892580343889328

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