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Properties

Label 2-770-55.32-c1-0-14
Degree $2$
Conductor $770$
Sign $0.842 - 0.538i$
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 + (−0.888 + 0.888i)3-s − 1.00i·4-s + (1.38 − 1.75i)5-s − 1.25i·6-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + 1.42i·9-s + (0.257 + 2.22i)10-s + (2.56 − 2.09i)11-s + (0.888 + 0.888i)12-s + (−1.35 − 1.35i)13-s − 1.00i·14-s + (0.323 + 2.79i)15-s − 1.00·16-s + (−2.87 + 2.87i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.512 + 0.512i)3-s − 0.500i·4-s + (0.620 − 0.783i)5-s − 0.512i·6-s + (−0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + 0.474i·9-s + (0.0815 + 0.702i)10-s + (0.774 − 0.632i)11-s + (0.256 + 0.256i)12-s + (−0.375 − 0.375i)13-s − 0.267i·14-s + (0.0836 + 0.720i)15-s − 0.250·16-s + (−0.698 + 0.698i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.842 - 0.538i$
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: \((2,\ 770,\ (\ :1/2),\ 0.842 - 0.538i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05687 + 0.308544i\)
\(L(\frac12)\) \(\approx\) \(1.05687 + 0.308544i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.38 + 1.75i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (-2.56 + 2.09i)T \)
good3 \( 1 + (0.888 - 0.888i)T - 3iT^{2} \)
13 \( 1 + (1.35 + 1.35i)T + 13iT^{2} \)
17 \( 1 + (2.87 - 2.87i)T - 17iT^{2} \)
19 \( 1 - 1.47T + 19T^{2} \)
23 \( 1 + (-4.72 + 4.72i)T - 23iT^{2} \)
29 \( 1 - 8.86T + 29T^{2} \)
31 \( 1 - 8.87T + 31T^{2} \)
37 \( 1 + (-1.13 - 1.13i)T + 37iT^{2} \)
41 \( 1 - 8.24iT - 41T^{2} \)
43 \( 1 + (4.20 + 4.20i)T + 43iT^{2} \)
47 \( 1 + (-8.69 - 8.69i)T + 47iT^{2} \)
53 \( 1 + (0.533 - 0.533i)T - 53iT^{2} \)
59 \( 1 + 9.93iT - 59T^{2} \)
61 \( 1 + 2.09iT - 61T^{2} \)
67 \( 1 + (-7.31 - 7.31i)T + 67iT^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 + (-10.4 - 10.4i)T + 73iT^{2} \)
79 \( 1 - 12.8T + 79T^{2} \)
83 \( 1 + (7.94 + 7.94i)T + 83iT^{2} \)
89 \( 1 - 8.22iT - 89T^{2} \)
97 \( 1 + (-1.76 - 1.76i)T + 97iT^{2} \)
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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

−10.21649441779181799111313292519, −9.577516877524674295428689900690, −8.626020867895005027156160986047, −8.167544904750062586318308174791, −6.62190010437078979690088830547, −6.08857737884803166689748198570, −5.06182127913132363674603718349, −4.43426656155517399211103142359, −2.60566352107871688097929153016, −0.965003911174125439398265888253, 1.03611065472115174117800989011, 2.36016257534697648086074331313, 3.47632483675586563393157896853, 4.78605874233171964788989240283, 6.15932340277273596501921804937, 6.92317371483057810602596298288, 7.27948070081117346766000293231, 8.844322214092483539409758485579, 9.543296273280676424452206353353, 10.16024858961128838488566297920

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