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Properties

Label 2-770-55.32-c1-0-24
Degree $2$
Conductor $770$
Sign $0.942 + 0.333i$
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 + (1.50 − 1.50i)3-s − 1.00i·4-s + (2.23 − 0.142i)5-s + 2.12i·6-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s − 1.51i·9-s + (−1.47 + 1.67i)10-s + (2.71 − 1.91i)11-s + (−1.50 − 1.50i)12-s + (0.722 + 0.722i)13-s − 1.00i·14-s + (3.13 − 3.56i)15-s − 1.00·16-s + (1.53 − 1.53i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.867 − 0.867i)3-s − 0.500i·4-s + (0.997 − 0.0638i)5-s + 0.867i·6-s + (−0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s − 0.505i·9-s + (−0.467 + 0.530i)10-s + (0.817 − 0.576i)11-s + (−0.433 − 0.433i)12-s + (0.200 + 0.200i)13-s − 0.267i·14-s + (0.810 − 0.921i)15-s − 0.250·16-s + (0.372 − 0.372i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.942 + 0.333i$
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: \((2,\ 770,\ (\ :1/2),\ 0.942 + 0.333i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90725 - 0.326922i\)
\(L(\frac12)\) \(\approx\) \(1.90725 - 0.326922i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 + 0.142i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (-2.71 + 1.91i)T \)
good3 \( 1 + (-1.50 + 1.50i)T - 3iT^{2} \)
13 \( 1 + (-0.722 - 0.722i)T + 13iT^{2} \)
17 \( 1 + (-1.53 + 1.53i)T - 17iT^{2} \)
19 \( 1 - 1.59T + 19T^{2} \)
23 \( 1 + (2.81 - 2.81i)T - 23iT^{2} \)
29 \( 1 - 1.98T + 29T^{2} \)
31 \( 1 + 6.13T + 31T^{2} \)
37 \( 1 + (2.22 + 2.22i)T + 37iT^{2} \)
41 \( 1 + 7.43iT - 41T^{2} \)
43 \( 1 + (-3.27 - 3.27i)T + 43iT^{2} \)
47 \( 1 + (4.35 + 4.35i)T + 47iT^{2} \)
53 \( 1 + (0.605 - 0.605i)T - 53iT^{2} \)
59 \( 1 - 1.98iT - 59T^{2} \)
61 \( 1 + 5.67iT - 61T^{2} \)
67 \( 1 + (2.05 + 2.05i)T + 67iT^{2} \)
71 \( 1 + 3.29T + 71T^{2} \)
73 \( 1 + (-3.33 - 3.33i)T + 73iT^{2} \)
79 \( 1 + 9.85T + 79T^{2} \)
83 \( 1 + (-5.34 - 5.34i)T + 83iT^{2} \)
89 \( 1 - 1.14iT - 89T^{2} \)
97 \( 1 + (-0.767 - 0.767i)T + 97iT^{2} \)
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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

−9.915670917334623895676792948236, −9.143559009277266054494302347239, −8.696917638520047373735346841848, −7.69016323217615078114871623400, −6.89882774634087181328020201119, −6.09642027621016231127723510515, −5.25468486896393874996810329756, −3.51058160956602278403957646213, −2.26992816840625524469351355098, −1.28810084676764577139199999373, 1.51746971612312694283103233456, 2.77321474936560665133316565795, 3.68380265783446372771106437492, 4.62986499273997464118538714999, 6.01043052188072304936972596027, 6.97843764059033389937324901500, 8.158724813821650691100266296965, 8.997935782772390009709380334259, 9.584521763427403692405985344083, 10.10942833151764595216704976140

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