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Properties

Label 2-770-35.27-c1-0-24
Degree $2$
Conductor $770$
Sign $0.775 - 0.631i$
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 + (2.03 + 2.03i)3-s + 1.00i·4-s + (2.07 + 0.842i)5-s − 2.88i·6-s + (0.703 − 2.55i)7-s + (0.707 − 0.707i)8-s + 5.31i·9-s + (−0.869 − 2.06i)10-s − 11-s + (−2.03 + 2.03i)12-s + (3.99 + 3.99i)13-s + (−2.30 + 1.30i)14-s + (2.50 + 5.93i)15-s − 1.00·16-s + (3.39 − 3.39i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (1.17 + 1.17i)3-s + 0.500i·4-s + (0.926 + 0.376i)5-s − 1.17i·6-s + (0.265 − 0.964i)7-s + (0.250 − 0.250i)8-s + 1.77i·9-s + (−0.274 − 0.651i)10-s − 0.301·11-s + (−0.588 + 0.588i)12-s + (1.10 + 1.10i)13-s + (−0.614 + 0.349i)14-s + (0.646 + 1.53i)15-s − 0.250·16-s + (0.823 − 0.823i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.775 - 0.631i$
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: \((2,\ 770,\ (\ :1/2),\ 0.775 - 0.631i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01140 + 0.715722i\)
\(L(\frac12)\) \(\approx\) \(2.01140 + 0.715722i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.07 - 0.842i)T \)
7 \( 1 + (-0.703 + 2.55i)T \)
11 \( 1 + T \)
good3 \( 1 + (-2.03 - 2.03i)T + 3iT^{2} \)
13 \( 1 + (-3.99 - 3.99i)T + 13iT^{2} \)
17 \( 1 + (-3.39 + 3.39i)T - 17iT^{2} \)
19 \( 1 + 4.95T + 19T^{2} \)
23 \( 1 + (2.50 - 2.50i)T - 23iT^{2} \)
29 \( 1 + 7.25iT - 29T^{2} \)
31 \( 1 + 3.87iT - 31T^{2} \)
37 \( 1 + (2.92 + 2.92i)T + 37iT^{2} \)
41 \( 1 - 4.22iT - 41T^{2} \)
43 \( 1 + (-1.04 + 1.04i)T - 43iT^{2} \)
47 \( 1 + (6.21 - 6.21i)T - 47iT^{2} \)
53 \( 1 + (0.793 - 0.793i)T - 53iT^{2} \)
59 \( 1 - 8.39T + 59T^{2} \)
61 \( 1 - 12.9iT - 61T^{2} \)
67 \( 1 + (-3.04 - 3.04i)T + 67iT^{2} \)
71 \( 1 + 9.47T + 71T^{2} \)
73 \( 1 + (8.93 + 8.93i)T + 73iT^{2} \)
79 \( 1 - 4.65iT - 79T^{2} \)
83 \( 1 + (10.6 + 10.6i)T + 83iT^{2} \)
89 \( 1 - 16.0T + 89T^{2} \)
97 \( 1 + (5.64 - 5.64i)T - 97iT^{2} \)
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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

−10.13761489557136150719444920777, −9.721180341308480805220987899950, −8.948127712911152907477847854960, −8.163325001494425194919158385808, −7.24508309341503335308464849457, −5.99682127951059209096865428076, −4.51109607677421435925532583653, −3.85196631245664534566023277110, −2.80899682659037926755852132573, −1.72501227839909537531802159757, 1.33685691243643484828954893737, 2.16361013430098476692163932544, 3.31489059532381498712577346727, 5.24265527238808668207148918645, 6.04856311780423263542877247693, 6.76168200217562480699468516491, 8.036667994496753439240965851686, 8.496892135784792353283071245807, 8.827687110545617237267507614081, 10.01489371895948555194118582123

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