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Properties

Label 2-770-55.32-c1-0-33
Degree $2$
Conductor $770$
Sign $-0.785 + 0.618i$
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 + (1.75 − 1.75i)3-s − 1.00i·4-s + (−0.321 − 2.21i)5-s + 2.48i·6-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s − 3.15i·9-s + (1.79 + 1.33i)10-s + (−3.31 − 0.100i)11-s + (−1.75 − 1.75i)12-s + (−4.93 − 4.93i)13-s − 1.00i·14-s + (−4.44 − 3.31i)15-s − 1.00·16-s + (−2.37 + 2.37i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (1.01 − 1.01i)3-s − 0.500i·4-s + (−0.143 − 0.989i)5-s + 1.01i·6-s + (−0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s − 1.05i·9-s + (0.566 + 0.422i)10-s + (−0.999 − 0.0303i)11-s + (−0.506 − 0.506i)12-s + (−1.36 − 1.36i)13-s − 0.267i·14-s + (−1.14 − 0.856i)15-s − 0.250·16-s + (−0.576 + 0.576i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 + 0.618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.785 + 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.785 + 0.618i$
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: \((2,\ 770,\ (\ :1/2),\ -0.785 + 0.618i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.319684 - 0.922209i\)
\(L(\frac12)\) \(\approx\) \(0.319684 - 0.922209i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.321 + 2.21i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (3.31 + 0.100i)T \)
good3 \( 1 + (-1.75 + 1.75i)T - 3iT^{2} \)
13 \( 1 + (4.93 + 4.93i)T + 13iT^{2} \)
17 \( 1 + (2.37 - 2.37i)T - 17iT^{2} \)
19 \( 1 + 0.0378T + 19T^{2} \)
23 \( 1 + (-1.66 + 1.66i)T - 23iT^{2} \)
29 \( 1 - 7.02T + 29T^{2} \)
31 \( 1 + 2.61T + 31T^{2} \)
37 \( 1 + (-6.70 - 6.70i)T + 37iT^{2} \)
41 \( 1 + 7.05iT - 41T^{2} \)
43 \( 1 + (3.13 + 3.13i)T + 43iT^{2} \)
47 \( 1 + (6.50 + 6.50i)T + 47iT^{2} \)
53 \( 1 + (-7.39 + 7.39i)T - 53iT^{2} \)
59 \( 1 - 4.17iT - 59T^{2} \)
61 \( 1 - 0.737iT - 61T^{2} \)
67 \( 1 + (3.59 + 3.59i)T + 67iT^{2} \)
71 \( 1 - 7.75T + 71T^{2} \)
73 \( 1 + (3.44 + 3.44i)T + 73iT^{2} \)
79 \( 1 - 5.31T + 79T^{2} \)
83 \( 1 + (4.10 + 4.10i)T + 83iT^{2} \)
89 \( 1 + 13.5iT - 89T^{2} \)
97 \( 1 + (8.40 + 8.40i)T + 97iT^{2} \)
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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

−9.794011624641648765451540663607, −8.685865747554398990124115109786, −8.293988320933978360269763649353, −7.64464299632753987765867448321, −6.82973871282763766586981731607, −5.58261350579957010362723250348, −4.77401449475226342713354892371, −3.00691906044899406567511914407, −2.05452316174725103913942315117, −0.47683773437330044195423367764, 2.43087805311314489615765810397, 2.87334285462749743962574884290, 4.05670258563211338494904553407, 4.82754907071417576585320001055, 6.61347807842903719458679408833, 7.42790278619944519293686370864, 8.203965765483477931350091784028, 9.457830846721158206639130177713, 9.573460400696741237849742376355, 10.47864177329621184746739518366

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