Properties

Label 2-770-385.164-c1-0-25
Degree $2$
Conductor $770$
Sign $0.329 + 0.944i$
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 + (0.730 − 1.26i)3-s + (−0.499 + 0.866i)4-s + (−0.672 + 2.13i)5-s − 1.46·6-s + (−1.48 − 2.18i)7-s + 0.999·8-s + (0.433 + 0.751i)9-s + (2.18 − 0.484i)10-s + (2.72 + 1.89i)11-s + (0.730 + 1.26i)12-s − 4.41i·13-s + (−1.15 + 2.38i)14-s + (2.20 + 2.40i)15-s + (−0.5 − 0.866i)16-s + (6.76 + 3.90i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.421 − 0.730i)3-s + (−0.249 + 0.433i)4-s + (−0.300 + 0.953i)5-s − 0.596·6-s + (−0.562 − 0.827i)7-s + 0.353·8-s + (0.144 + 0.250i)9-s + (0.690 − 0.153i)10-s + (0.821 + 0.569i)11-s + (0.210 + 0.365i)12-s − 1.22i·13-s + (−0.307 + 0.636i)14-s + (0.569 + 0.621i)15-s + (−0.125 − 0.216i)16-s + (1.64 + 0.946i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.329 + 0.944i$
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: \((2,\ 770,\ (\ :1/2),\ 0.329 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14443 - 0.813042i\)
\(L(\frac12)\) \(\approx\) \(1.14443 - 0.813042i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.672 - 2.13i)T \)
7 \( 1 + (1.48 + 2.18i)T \)
11 \( 1 + (-2.72 - 1.89i)T \)
good3 \( 1 + (-0.730 + 1.26i)T + (-1.5 - 2.59i)T^{2} \)
13 \( 1 + 4.41iT - 13T^{2} \)
17 \( 1 + (-6.76 - 3.90i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.986 + 1.70i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.107 + 0.0621i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.04iT - 29T^{2} \)
31 \( 1 + (-2.78 - 1.60i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-9.23 + 5.33i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.22T + 41T^{2} \)
43 \( 1 - 4.97T + 43T^{2} \)
47 \( 1 + (-1.30 - 2.26i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.19 + 4.73i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.46 - 4.88i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.840 + 1.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.21 - 1.27i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.5T + 71T^{2} \)
73 \( 1 + (8.48 + 4.89i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.18 - 3.57i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.51iT - 83T^{2} \)
89 \( 1 + (0.291 - 0.168i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.4T + 97T^{2} \)
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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

−10.18191806337136304265761079434, −9.623108129590252221555806327068, −8.081141638317583781577904540608, −7.76547462871288247832147247814, −6.93876056291716911857058046350, −6.00162909679588139358858595524, −4.25017263316293354669237703997, −3.38704962790443863381542153964, −2.41983243479368196801381011425, −0.987321555088788120776066865190, 1.20461897622617908621788130866, 3.18930756475101945178847929654, 4.15979665964581006576122371096, 5.12746696832322316122164828738, 6.08819918547687738054773529102, 7.01446934665085845412153105836, 8.234474064831704777746352963825, 8.889933930215427061169170146080, 9.480659184750185789950752078558, 9.891343730776706849203864059846

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