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Properties

Label 2-770-35.27-c1-0-15
Degree $2$
Conductor $770$
Sign $-0.511 - 0.859i$
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 + (2.23 + 2.23i)3-s + 1.00i·4-s + (0.288 − 2.21i)5-s + 3.15i·6-s + (−1.64 + 2.06i)7-s + (−0.707 + 0.707i)8-s + 6.95i·9-s + (1.77 − 1.36i)10-s − 11-s + (−2.23 + 2.23i)12-s + (2.34 + 2.34i)13-s + (−2.62 + 0.298i)14-s + (5.59 − 4.30i)15-s − 1.00·16-s + (3.89 − 3.89i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (1.28 + 1.28i)3-s + 0.500i·4-s + (0.128 − 0.991i)5-s + 1.28i·6-s + (−0.622 + 0.782i)7-s + (−0.250 + 0.250i)8-s + 2.31i·9-s + (0.560 − 0.431i)10-s − 0.301·11-s + (−0.644 + 0.644i)12-s + (0.650 + 0.650i)13-s + (−0.702 + 0.0797i)14-s + (1.44 − 1.11i)15-s − 0.250·16-s + (0.944 − 0.944i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.511 - 0.859i$
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: \((2,\ 770,\ (\ :1/2),\ -0.511 - 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41234 + 2.48558i\)
\(L(\frac12)\) \(\approx\) \(1.41234 + 2.48558i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.288 + 2.21i)T \)
7 \( 1 + (1.64 - 2.06i)T \)
11 \( 1 + T \)
good3 \( 1 + (-2.23 - 2.23i)T + 3iT^{2} \)
13 \( 1 + (-2.34 - 2.34i)T + 13iT^{2} \)
17 \( 1 + (-3.89 + 3.89i)T - 17iT^{2} \)
19 \( 1 + 1.89T + 19T^{2} \)
23 \( 1 + (0.0242 - 0.0242i)T - 23iT^{2} \)
29 \( 1 - 8.09iT - 29T^{2} \)
31 \( 1 + 8.61iT - 31T^{2} \)
37 \( 1 + (-6.66 - 6.66i)T + 37iT^{2} \)
41 \( 1 + 11.7iT - 41T^{2} \)
43 \( 1 + (1.62 - 1.62i)T - 43iT^{2} \)
47 \( 1 + (-5.09 + 5.09i)T - 47iT^{2} \)
53 \( 1 + (-6.27 + 6.27i)T - 53iT^{2} \)
59 \( 1 - 0.608T + 59T^{2} \)
61 \( 1 - 10.2iT - 61T^{2} \)
67 \( 1 + (5.29 + 5.29i)T + 67iT^{2} \)
71 \( 1 + 4.98T + 71T^{2} \)
73 \( 1 + (-2.98 - 2.98i)T + 73iT^{2} \)
79 \( 1 - 5.09iT - 79T^{2} \)
83 \( 1 + (4.36 + 4.36i)T + 83iT^{2} \)
89 \( 1 - 18.5T + 89T^{2} \)
97 \( 1 + (-6.57 + 6.57i)T - 97iT^{2} \)
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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

−10.21614558793787102845370805900, −9.486940407226916304023644022913, −8.857873752058643939746288738903, −8.388745386088436858716064282928, −7.33100058889531470538289615187, −5.84501495105825669360451806442, −5.10818556118085132542268798271, −4.20823237603808212998628771922, −3.35528174480988604481503784608, −2.30291750722612480638081983280, 1.16329395109999196179429616882, 2.47298251697771898375609672454, 3.24201713635951126265447495061, 3.93772071505747313027481110563, 6.01856656607811313353088773628, 6.48278031532106915775680089946, 7.56592901648993472833788129359, 8.016989129433756475856240487625, 9.229457371679078064181745315497, 10.16789739800580323105349065916

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