↑

Properties

Label 2-770-55.32-c1-0-35
Degree $2$
Conductor $770$
Sign $-0.838 + 0.545i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s + (2 − 2i)3-s − 1.00i·4-s + (−1 − 2i)5-s − 2.82i·6-s + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s − 5i·9-s + (−2.12 − 0.707i)10-s + (3 + 1.41i)11-s + (−2.00 − 2.00i)12-s + (−1.41 − 1.41i)13-s + 1.00i·14-s + (−6 − 2i)15-s − 1.00·16-s + (−1.41 + 1.41i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (1.15 − 1.15i)3-s − 0.500i·4-s + (−0.447 − 0.894i)5-s − 1.15i·6-s + (−0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s − 1.66i·9-s + (−0.670 − 0.223i)10-s + (0.904 + 0.426i)11-s + (−0.577 − 0.577i)12-s + (−0.392 − 0.392i)13-s + 0.267i·14-s + (−1.54 − 0.516i)15-s − 0.250·16-s + (−0.342 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 + 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.838 + 0.545i$
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: \((2,\ 770,\ (\ :1/2),\ -0.838 + 0.545i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.740672 - 2.49701i\)
\(L(\frac12)\) \(\approx\) \(0.740672 - 2.49701i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 2i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (-3 - 1.41i)T \)
good3 \( 1 + (-2 + 2i)T - 3iT^{2} \)
13 \( 1 + (1.41 + 1.41i)T + 13iT^{2} \)
17 \( 1 + (1.41 - 1.41i)T - 17iT^{2} \)
19 \( 1 - 4.24T + 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + 1.41T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (4 + 4i)T + 37iT^{2} \)
41 \( 1 + 4.24iT - 41T^{2} \)
43 \( 1 + (-5.65 - 5.65i)T + 43iT^{2} \)
47 \( 1 + (-5 - 5i)T + 47iT^{2} \)
53 \( 1 + (6 - 6i)T - 53iT^{2} \)
59 \( 1 + 8iT - 59T^{2} \)
61 \( 1 + 5.65iT - 61T^{2} \)
67 \( 1 + (-3 - 3i)T + 67iT^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (-11.3 - 11.3i)T + 73iT^{2} \)
79 \( 1 - 7.07T + 79T^{2} \)
83 \( 1 + (7.07 + 7.07i)T + 83iT^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + (-2 - 2i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.572774692885775341375070574850, −9.192978307328717106485639863377, −8.240483378422437883652158196941, −7.48263829700447805087406912619, −6.59751100078320851016459118319, −5.42890308659550983019848431708, −4.20002169759427169217719176419, −3.26849221659852822896550991430, −2.13192332177002063512102761674, −1.06187127035623426419716766658, 2.56700738414889156815272863635, 3.49152382259985417621346215858, 4.00941523822253993473325558394, 5.04858323021295868530515840936, 6.42467303392393843388064657247, 7.23048649971555986425190608223, 8.083488445494084341183760079331, 9.020295326564210171054111778194, 9.656152708382050123247362082708, 10.53068666059685779802282745621

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