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Properties

Label 2-770-35.27-c1-0-23
Degree $2$
Conductor $770$
Sign $-0.417 - 0.908i$
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 + (1.49 + 1.49i)3-s + 1.00i·4-s + (0.446 + 2.19i)5-s + 2.11i·6-s + (2.36 − 1.17i)7-s + (−0.707 + 0.707i)8-s + 1.47i·9-s + (−1.23 + 1.86i)10-s − 11-s + (−1.49 + 1.49i)12-s + (0.153 + 0.153i)13-s + (2.50 + 0.842i)14-s + (−2.61 + 3.94i)15-s − 1.00·16-s + (−0.185 + 0.185i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.863 + 0.863i)3-s + 0.500i·4-s + (0.199 + 0.979i)5-s + 0.863i·6-s + (0.895 − 0.444i)7-s + (−0.250 + 0.250i)8-s + 0.491i·9-s + (−0.390 + 0.589i)10-s − 0.301·11-s + (−0.431 + 0.431i)12-s + (0.0426 + 0.0426i)13-s + (0.670 + 0.225i)14-s + (−0.673 + 1.01i)15-s − 0.250·16-s + (−0.0449 + 0.0449i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.417 - 0.908i$
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: \((2,\ 770,\ (\ :1/2),\ -0.417 - 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55202 + 2.41998i\)
\(L(\frac12)\) \(\approx\) \(1.55202 + 2.41998i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.446 - 2.19i)T \)
7 \( 1 + (-2.36 + 1.17i)T \)
11 \( 1 + T \)
good3 \( 1 + (-1.49 - 1.49i)T + 3iT^{2} \)
13 \( 1 + (-0.153 - 0.153i)T + 13iT^{2} \)
17 \( 1 + (0.185 - 0.185i)T - 17iT^{2} \)
19 \( 1 - 3.14T + 19T^{2} \)
23 \( 1 + (2.00 - 2.00i)T - 23iT^{2} \)
29 \( 1 - 1.96iT - 29T^{2} \)
31 \( 1 + 8.77iT - 31T^{2} \)
37 \( 1 + (4.87 + 4.87i)T + 37iT^{2} \)
41 \( 1 + 7.00iT - 41T^{2} \)
43 \( 1 + (4.12 - 4.12i)T - 43iT^{2} \)
47 \( 1 + (-2.57 + 2.57i)T - 47iT^{2} \)
53 \( 1 + (4.52 - 4.52i)T - 53iT^{2} \)
59 \( 1 + 7.31T + 59T^{2} \)
61 \( 1 - 8.14iT - 61T^{2} \)
67 \( 1 + (2.91 + 2.91i)T + 67iT^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 + (-3.13 - 3.13i)T + 73iT^{2} \)
79 \( 1 + 11.0iT - 79T^{2} \)
83 \( 1 + (-8.41 - 8.41i)T + 83iT^{2} \)
89 \( 1 + 2.17T + 89T^{2} \)
97 \( 1 + (-12.3 + 12.3i)T - 97iT^{2} \)
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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

−10.52401698122351082164357082951, −9.727726806485759366526619299965, −8.862775411372103142339773208947, −7.83597715793933976124064323126, −7.32974336015315689726905593229, −6.13486042123627097960731926820, −5.11679928593904349533699728516, −4.05929702740341215897223191128, −3.37071972148191783716625167120, −2.21958967604629032768285273781, 1.31097338017325054235950871072, 2.11615438422674508199297995407, 3.25383809729576838480743372892, 4.72564385167274296203475304643, 5.27857386291548699782510939939, 6.50828694397406043730192149769, 7.73153430467403386914587427778, 8.343365982483996706866662005179, 9.011380785243245252232295659713, 10.00716945381266705647008571052

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