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Properties

Label 2-770-35.13-c1-0-11
Degree $2$
Conductor $770$
Sign $0.110 - 0.993i$
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 + (−0.903 + 0.903i)3-s − 1.00i·4-s + (−1.41 − 1.72i)5-s − 1.27i·6-s + (1.36 + 2.26i)7-s + (0.707 + 0.707i)8-s + 1.36i·9-s + (2.22 + 0.221i)10-s − 11-s + (0.903 + 0.903i)12-s + (4.51 − 4.51i)13-s + (−2.56 − 0.638i)14-s + (2.84 + 0.283i)15-s − 1.00·16-s + (−4.77 − 4.77i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.521 + 0.521i)3-s − 0.500i·4-s + (−0.633 − 0.773i)5-s − 0.521i·6-s + (0.515 + 0.856i)7-s + (0.250 + 0.250i)8-s + 0.455i·9-s + (0.703 + 0.0700i)10-s − 0.301·11-s + (0.260 + 0.260i)12-s + (1.25 − 1.25i)13-s + (−0.686 − 0.170i)14-s + (0.734 + 0.0730i)15-s − 0.250·16-s + (−1.15 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.110 - 0.993i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (573, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.110 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.652345 + 0.583598i\)
\(L(\frac12)\) \(\approx\) \(0.652345 + 0.583598i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.41 + 1.72i)T \)
7 \( 1 + (-1.36 - 2.26i)T \)
11 \( 1 + T \)
good3 \( 1 + (0.903 - 0.903i)T - 3iT^{2} \)
13 \( 1 + (-4.51 + 4.51i)T - 13iT^{2} \)
17 \( 1 + (4.77 + 4.77i)T + 17iT^{2} \)
19 \( 1 - 6.06T + 19T^{2} \)
23 \( 1 + (-1.28 - 1.28i)T + 23iT^{2} \)
29 \( 1 - 7.55iT - 29T^{2} \)
31 \( 1 - 7.78iT - 31T^{2} \)
37 \( 1 + (-4.82 + 4.82i)T - 37iT^{2} \)
41 \( 1 - 5.27iT - 41T^{2} \)
43 \( 1 + (-6.22 - 6.22i)T + 43iT^{2} \)
47 \( 1 + (-4.65 - 4.65i)T + 47iT^{2} \)
53 \( 1 + (-4.24 - 4.24i)T + 53iT^{2} \)
59 \( 1 - 1.55T + 59T^{2} \)
61 \( 1 + 0.913iT - 61T^{2} \)
67 \( 1 + (8.57 - 8.57i)T - 67iT^{2} \)
71 \( 1 - 6.02T + 71T^{2} \)
73 \( 1 + (2.79 - 2.79i)T - 73iT^{2} \)
79 \( 1 + 6.11iT - 79T^{2} \)
83 \( 1 + (-5.04 + 5.04i)T - 83iT^{2} \)
89 \( 1 - 7.76T + 89T^{2} \)
97 \( 1 + (6.01 + 6.01i)T + 97iT^{2} \)
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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

−10.70647913557045576953326953515, −9.391156306732164042802038124066, −8.825382013202978188594130983682, −8.014084131686953964134561212817, −7.31321371724483856812645713292, −5.80254597081616085138831507892, −5.25272208786119282980762141176, −4.57396054286591606185951083111, −2.97370967962627792068756049760, −1.10120176614974865990033663033, 0.71778685445843379647754327620, 2.08469680796144944517826974254, 3.72558092162722515276325305668, 4.21390056105672171316810917719, 6.02898015099829897463150169252, 6.77363556101704286444798041215, 7.50911740770077626389469415454, 8.337585197124521817241975388118, 9.319585158402893467493933826250, 10.37888621314973261252494706338

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