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Properties

Label 2-770-11.3-c1-0-16
Degree $2$
Conductor $770$
Sign $-0.546 + 0.837i$
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  + (−0.809 + 0.587i)2-s + (−0.248 − 0.764i)3-s + (0.309 − 0.951i)4-s + (−0.809 − 0.587i)5-s + (0.649 + 0.472i)6-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (1.90 − 1.38i)9-s + 10-s + (−3.22 − 0.787i)11-s − 0.803·12-s + (2.13 − 1.55i)13-s + (−0.309 − 0.951i)14-s + (−0.248 + 0.764i)15-s + (−0.809 − 0.587i)16-s + (−1.29 − 0.943i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.143 − 0.441i)3-s + (0.154 − 0.475i)4-s + (−0.361 − 0.262i)5-s + (0.265 + 0.192i)6-s + (−0.116 + 0.359i)7-s + (0.109 + 0.336i)8-s + (0.634 − 0.461i)9-s + 0.316·10-s + (−0.971 − 0.237i)11-s − 0.231·12-s + (0.593 − 0.430i)13-s + (−0.0825 − 0.254i)14-s + (−0.0641 + 0.197i)15-s + (−0.202 − 0.146i)16-s + (−0.314 − 0.228i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.546 + 0.837i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.546 + 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.268449 - 0.495516i\)
\(L(\frac12)\) \(\approx\) \(0.268449 - 0.495516i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (0.809 + 0.587i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (3.22 + 0.787i)T \)
good3 \( 1 + (0.248 + 0.764i)T + (-2.42 + 1.76i)T^{2} \)
13 \( 1 + (-2.13 + 1.55i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.29 + 0.943i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.590 - 1.81i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 4.22T + 23T^{2} \)
29 \( 1 + (-2.77 + 8.53i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.00 - 1.45i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.73 - 5.33i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.99 + 9.22i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 4.00T + 43T^{2} \)
47 \( 1 + (2.02 + 6.22i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (8.12 - 5.90i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.28 + 3.96i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (10.7 + 7.79i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 + (3.70 + 2.69i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.14 + 3.51i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-11.2 + 8.19i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-10.9 - 7.97i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 3.41T + 89T^{2} \)
97 \( 1 + (-14.1 + 10.2i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.00149819362455504154633216176, −9.064506399845392821183642346727, −8.126498434304878032208166714094, −7.65913908879992353355073704403, −6.53477761766427191810775447109, −5.85062345303281059651685563229, −4.75100165017523418467865878767, −3.42098366355059952679386687285, −1.88741011747581955470639903396, −0.34700528466490223138013440953, 1.67803649767918143725619313730, 3.07345953164706073192284151449, 4.14403331171385315555888576977, 5.00665302024096211039325689714, 6.42514268350975167434983240893, 7.36558298600824013054018835231, 8.036983384750887433571668022659, 9.045603028295012499078034062012, 9.939515402813418303724739508854, 10.64832353630089841734944689869

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