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Properties

Label 2-770-11.4-c1-0-5
Degree $2$
Conductor $770$
Sign $-0.991 - 0.130i$
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 + (−0.521 + 1.60i)3-s + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s + (−1.36 + 0.991i)6-s + (0.309 + 0.951i)7-s + (−0.309 + 0.951i)8-s + (0.124 + 0.0906i)9-s − 10-s + (−1.49 + 2.95i)11-s − 1.68·12-s + (−0.0344 − 0.0250i)13-s + (−0.309 + 0.951i)14-s + (−0.521 − 1.60i)15-s + (−0.809 + 0.587i)16-s + (3.81 − 2.77i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−0.300 + 0.926i)3-s + (0.154 + 0.475i)4-s + (−0.361 + 0.262i)5-s + (−0.557 + 0.404i)6-s + (0.116 + 0.359i)7-s + (−0.109 + 0.336i)8-s + (0.0416 + 0.0302i)9-s − 0.316·10-s + (−0.451 + 0.892i)11-s − 0.486·12-s + (−0.00955 − 0.00694i)13-s + (−0.0825 + 0.254i)14-s + (−0.134 − 0.414i)15-s + (−0.202 + 0.146i)16-s + (0.926 − 0.672i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.991 - 0.130i$
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: \((2,\ 770,\ (\ :1/2),\ -0.991 - 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0988865 + 1.51201i\)
\(L(\frac12)\) \(\approx\) \(0.0988865 + 1.51201i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.49 - 2.95i)T \)
good3 \( 1 + (0.521 - 1.60i)T + (-2.42 - 1.76i)T^{2} \)
13 \( 1 + (0.0344 + 0.0250i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.81 + 2.77i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.529 - 1.62i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 6.95T + 23T^{2} \)
29 \( 1 + (0.414 + 1.27i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.86 + 3.53i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.62 - 8.06i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.20 + 3.72i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 6.42T + 43T^{2} \)
47 \( 1 + (-1.41 + 4.36i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-11.4 - 8.28i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.15 + 9.71i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (11.2 - 8.15i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 6.39T + 67T^{2} \)
71 \( 1 + (9.14 - 6.64i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.614 + 1.89i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-7.17 - 5.21i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-8.15 + 5.92i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 2.93T + 89T^{2} \)
97 \( 1 + (-14.1 - 10.2i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.59402043336968451608389116583, −10.02125096866675582193594055371, −9.147582575615315127143019753194, −7.82715014016688539153898268678, −7.42189181868811208247047662637, −6.08109374428605937379061139839, −5.28459068940036753346191661209, −4.44168636917960165133842843496, −3.65305274037665095438449585568, −2.28610606518893362117725516724, 0.65832266507572345647934388844, 1.90536624329982936118739202942, 3.38987808916570256324063870935, 4.28770763519240524675574867209, 5.59924201867990613761064309997, 6.18112905226248781752478010991, 7.36090636459456028690521298465, 7.929430198091832018601417009601, 9.045828575597354196469606093038, 10.19783293285474332384717172076

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