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Properties

Label 2-770-11.3-c1-0-7
Degree $2$
Conductor $770$
Sign $0.860 + 0.508i$
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 + (−0.563 − 1.73i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (1.47 + 1.07i)6-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (−0.265 + 0.192i)9-s − 10-s + (−0.645 + 3.25i)11-s − 1.82·12-s + (1.79 − 1.30i)13-s + (0.309 + 0.951i)14-s + (0.563 − 1.73i)15-s + (−0.809 − 0.587i)16-s + (6.44 + 4.68i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.325 − 1.00i)3-s + (0.154 − 0.475i)4-s + (0.361 + 0.262i)5-s + (0.602 + 0.437i)6-s + (0.116 − 0.359i)7-s + (0.109 + 0.336i)8-s + (−0.0885 + 0.0643i)9-s − 0.316·10-s + (−0.194 + 0.980i)11-s − 0.526·12-s + (0.496 − 0.360i)13-s + (0.0825 + 0.254i)14-s + (0.145 − 0.447i)15-s + (−0.202 − 0.146i)16-s + (1.56 + 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.860 + 0.508i$
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: \((2,\ 770,\ (\ :1/2),\ 0.860 + 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15857 - 0.316755i\)
\(L(\frac12)\) \(\approx\) \(1.15857 - 0.316755i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.645 - 3.25i)T \)
good3 \( 1 + (0.563 + 1.73i)T + (-2.42 + 1.76i)T^{2} \)
13 \( 1 + (-1.79 + 1.30i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-6.44 - 4.68i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.282 + 0.870i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 6.07T + 23T^{2} \)
29 \( 1 + (-1.56 + 4.81i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (7.55 - 5.48i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.33 + 4.11i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.78 + 8.55i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 2.76T + 43T^{2} \)
47 \( 1 + (-1.30 - 4.02i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-7.55 + 5.48i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.22 + 6.84i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-10.2 - 7.45i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 1.29T + 67T^{2} \)
71 \( 1 + (0.0163 + 0.0118i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.66 + 11.2i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-7.59 + 5.52i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.604 + 0.439i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 14.2T + 89T^{2} \)
97 \( 1 + (-5.88 + 4.27i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.28406121770160717925133708218, −9.402080240935582711198489334958, −8.328605352303578074139254696787, −7.42969089956977949549531332471, −7.01659972290362017822931910578, −6.03281047606277080026987110620, −5.25885798884002448217816745102, −3.70430618904254656635599658776, −2.04671297592137432021486251556, −1.00975620991228537046404803118, 1.15910193692239697504518769756, 2.85041684210329037609721673882, 3.81987274013234044346272611630, 5.10645257416206366315926812360, 5.65943097824290808444890743334, 7.02217102143773456888851705536, 8.095167785502507545922252123843, 8.991377381325425219725818860976, 9.580837494967958644493093114392, 10.31312011586934899753651209396

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