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Properties

Label 2-770-11.4-c1-0-1
Degree $2$
Conductor $770$
Sign $-0.902 + 0.430i$
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  + (−0.809 − 0.587i)2-s + (−0.912 + 2.80i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (2.38 − 1.73i)6-s + (0.309 + 0.951i)7-s + (0.309 − 0.951i)8-s + (−4.62 − 3.35i)9-s − 10-s + (0.355 + 3.29i)11-s − 2.95·12-s + (−4.69 − 3.40i)13-s + (0.309 − 0.951i)14-s + (0.912 + 2.80i)15-s + (−0.809 + 0.587i)16-s + (0.380 − 0.276i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.526 + 1.62i)3-s + (0.154 + 0.475i)4-s + (0.361 − 0.262i)5-s + (0.974 − 0.708i)6-s + (0.116 + 0.359i)7-s + (0.109 − 0.336i)8-s + (−1.54 − 1.11i)9-s − 0.316·10-s + (0.107 + 0.994i)11-s − 0.852·12-s + (−1.30 − 0.945i)13-s + (0.0825 − 0.254i)14-s + (0.235 + 0.724i)15-s + (−0.202 + 0.146i)16-s + (0.0921 − 0.0669i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.902 + 0.430i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.902 + 0.430i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0626549 - 0.276668i\)
\(L(\frac12)\) \(\approx\) \(0.0626549 - 0.276668i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.355 - 3.29i)T \)
good3 \( 1 + (0.912 - 2.80i)T + (-2.42 - 1.76i)T^{2} \)
13 \( 1 + (4.69 + 3.40i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.380 + 0.276i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.25 - 3.84i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 7.69T + 23T^{2} \)
29 \( 1 + (-0.813 - 2.50i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.60 - 1.89i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.40 + 4.33i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.563 + 1.73i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 0.790T + 43T^{2} \)
47 \( 1 + (1.32 - 4.07i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (11.2 + 8.19i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.02 + 12.3i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-5.41 + 3.93i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 6.73T + 67T^{2} \)
71 \( 1 + (9.29 - 6.75i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.78 - 11.6i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (3.19 + 2.32i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-7.64 + 5.55i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + (-13.5 - 9.84i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.41785094428358224231079039352, −9.935372727594131258066481841170, −9.588857174114940338136641461699, −8.530849318256566875487532215582, −7.62364339481962150886775241407, −6.19891789633633115035565490492, −5.22022721419420648860763993984, −4.54467187898185906052406241539, −3.44515534510239774431643055607, −2.12365801311045878641704894090, 0.17603019498076785973661838631, 1.60377925463916146990874669635, 2.62852497017278568599346188663, 4.64111962974779014232253089419, 5.93714863987025562193287671229, 6.39876345010616620158318574118, 7.24310112430805155228790882553, 7.84165652820803654914603942239, 8.756527566738738973635731991666, 9.799519832601130830192065761820

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