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Properties

Label 2-770-35.13-c1-0-9
Degree $2$
Conductor $770$
Sign $0.939 - 0.341i$
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 + (−2.03 + 2.03i)3-s − 1.00i·4-s + (−2.07 + 0.842i)5-s − 2.88i·6-s + (−2.55 − 0.703i)7-s + (0.707 + 0.707i)8-s − 5.31i·9-s + (0.869 − 2.06i)10-s − 11-s + (2.03 + 2.03i)12-s + (−3.99 + 3.99i)13-s + (2.30 − 1.30i)14-s + (2.50 − 5.93i)15-s − 1.00·16-s + (−3.39 − 3.39i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−1.17 + 1.17i)3-s − 0.500i·4-s + (−0.926 + 0.376i)5-s − 1.17i·6-s + (−0.964 − 0.265i)7-s + (0.250 + 0.250i)8-s − 1.77i·9-s + (0.274 − 0.651i)10-s − 0.301·11-s + (0.588 + 0.588i)12-s + (−1.10 + 1.10i)13-s + (0.614 − 0.349i)14-s + (0.646 − 1.53i)15-s − 0.250·16-s + (−0.823 − 0.823i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.939 - 0.341i$
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: \((2,\ 770,\ (\ :1/2),\ 0.939 - 0.341i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.196292 + 0.0345989i\)
\(L(\frac12)\) \(\approx\) \(0.196292 + 0.0345989i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.07 - 0.842i)T \)
7 \( 1 + (2.55 + 0.703i)T \)
11 \( 1 + T \)
good3 \( 1 + (2.03 - 2.03i)T - 3iT^{2} \)
13 \( 1 + (3.99 - 3.99i)T - 13iT^{2} \)
17 \( 1 + (3.39 + 3.39i)T + 17iT^{2} \)
19 \( 1 - 4.95T + 19T^{2} \)
23 \( 1 + (2.50 + 2.50i)T + 23iT^{2} \)
29 \( 1 - 7.25iT - 29T^{2} \)
31 \( 1 + 3.87iT - 31T^{2} \)
37 \( 1 + (2.92 - 2.92i)T - 37iT^{2} \)
41 \( 1 - 4.22iT - 41T^{2} \)
43 \( 1 + (-1.04 - 1.04i)T + 43iT^{2} \)
47 \( 1 + (-6.21 - 6.21i)T + 47iT^{2} \)
53 \( 1 + (0.793 + 0.793i)T + 53iT^{2} \)
59 \( 1 + 8.39T + 59T^{2} \)
61 \( 1 - 12.9iT - 61T^{2} \)
67 \( 1 + (-3.04 + 3.04i)T - 67iT^{2} \)
71 \( 1 + 9.47T + 71T^{2} \)
73 \( 1 + (-8.93 + 8.93i)T - 73iT^{2} \)
79 \( 1 + 4.65iT - 79T^{2} \)
83 \( 1 + (-10.6 + 10.6i)T - 83iT^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
97 \( 1 + (-5.64 - 5.64i)T + 97iT^{2} \)
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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

−10.31344827006483821978388515231, −9.592522382830802647925058459674, −9.005260205628152337314608764733, −7.48527683304633101946389699205, −6.90563473017087543505124972536, −6.05108570641423291266095687190, −4.86649260422198575700640012219, −4.30649863278694364991228391979, −3.01571991957318402892534289883, −0.23762110073499565959066087237, 0.67242784242197516196363748216, 2.29458879310514156047781004916, 3.61116490748032485684272617403, 5.07232015580240402242542359993, 5.90932800111394439074783478879, 7.03073003199918723857418623744, 7.58203828098971872043539058563, 8.374092844257987364935468856912, 9.563639894500852549380358156454, 10.45092878572606671409473034190

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