↑

Properties

Label 2-770-11.3-c1-0-6
Degree $2$
Conductor $770$
Sign $0.205 - 0.978i$
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  + (0.809 − 0.587i)2-s + (0.941 + 2.89i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (2.46 + 1.79i)6-s + (−0.309 + 0.951i)7-s + (−0.309 − 0.951i)8-s + (−5.08 + 3.69i)9-s + 10-s + (3.28 − 0.432i)11-s + 3.04·12-s + (0.285 − 0.207i)13-s + (0.309 + 0.951i)14-s + (−0.941 + 2.89i)15-s + (−0.809 − 0.587i)16-s + (0.655 + 0.476i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.543 + 1.67i)3-s + (0.154 − 0.475i)4-s + (0.361 + 0.262i)5-s + (1.00 + 0.731i)6-s + (−0.116 + 0.359i)7-s + (−0.109 − 0.336i)8-s + (−1.69 + 1.23i)9-s + 0.316·10-s + (0.991 − 0.130i)11-s + 0.879·12-s + (0.0792 − 0.0575i)13-s + (0.0825 + 0.254i)14-s + (−0.243 + 0.748i)15-s + (−0.202 − 0.146i)16-s + (0.159 + 0.115i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.205 - 0.978i$
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: \((2,\ 770,\ (\ :1/2),\ 0.205 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02975 + 1.64726i\)
\(L(\frac12)\) \(\approx\) \(2.02975 + 1.64726i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.28 + 0.432i)T \)
good3 \( 1 + (-0.941 - 2.89i)T + (-2.42 + 1.76i)T^{2} \)
13 \( 1 + (-0.285 + 0.207i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.655 - 0.476i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.801 - 2.46i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 2T + 23T^{2} \)
29 \( 1 + (2.95 - 9.08i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.61 + 3.35i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.10 + 3.38i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.55 + 7.85i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 9.49T + 43T^{2} \)
47 \( 1 + (2.65 + 8.18i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.26 + 1.64i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-4.08 + 12.5i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-5.42 - 3.94i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 1.40T + 67T^{2} \)
71 \( 1 + (-10.4 - 7.59i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.59 - 4.89i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.95 + 2.14i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.15 - 1.56i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 9.63T + 89T^{2} \)
97 \( 1 + (-7.92 + 5.75i)T + (29.9 - 92.2i)T^{2} \)
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

−10.31618973010560778750703215518, −9.852973480555509946584093272429, −9.060790285012672216079751090367, −8.351141598688486300107670568481, −6.79406625349030129816434625721, −5.69183696726637058655883645391, −4.99179061415866599191073416086, −3.80324666017818163036027095528, −3.39144639849295473779361998480, −2.06849148666818694313496952581, 1.13531311212004966933531085778, 2.32565971314654979214787992994, 3.47792550817614661042012192962, 4.76891626280436564411493266458, 6.21530421462477562720388336715, 6.48665563830501679271678286459, 7.47146402934863226198226622366, 8.136308419624428942579619683321, 8.993804080387226977825381212264, 9.927981757749735013123274701604

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