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Properties

Label 2-770-11.3-c1-0-21
Degree $2$
Conductor $770$
Sign $-0.587 - 0.809i$
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 + (−0.867 − 2.67i)3-s + (0.309 − 0.951i)4-s + (−0.809 − 0.587i)5-s + (2.27 + 1.65i)6-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (−3.95 + 2.87i)9-s + 10-s + (−1.88 + 2.72i)11-s − 2.80·12-s + (2.47 − 1.79i)13-s + (0.309 + 0.951i)14-s + (−0.867 + 2.67i)15-s + (−0.809 − 0.587i)16-s + (−5.61 − 4.08i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (−0.501 − 1.54i)3-s + (0.154 − 0.475i)4-s + (−0.361 − 0.262i)5-s + (0.927 + 0.673i)6-s + (0.116 − 0.359i)7-s + (0.109 + 0.336i)8-s + (−1.31 + 0.957i)9-s + 0.316·10-s + (−0.569 + 0.822i)11-s − 0.810·12-s + (0.686 − 0.498i)13-s + (0.0825 + 0.254i)14-s + (−0.224 + 0.689i)15-s + (−0.202 − 0.146i)16-s + (−1.36 − 0.990i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.587 - 0.809i$
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: \((2,\ 770,\ (\ :1/2),\ -0.587 - 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0629751 + 0.123448i\)
\(L(\frac12)\) \(\approx\) \(0.0629751 + 0.123448i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.88 - 2.72i)T \)
good3 \( 1 + (0.867 + 2.67i)T + (-2.42 + 1.76i)T^{2} \)
13 \( 1 + (-2.47 + 1.79i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (5.61 + 4.08i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.45 + 4.46i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.73T + 23T^{2} \)
29 \( 1 + (2.08 - 6.41i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.70 - 1.96i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.26 - 6.96i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.34 - 7.21i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.27T + 43T^{2} \)
47 \( 1 + (-1.23 - 3.80i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.37 + 0.999i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.36 + 7.27i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (7.22 + 5.24i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 10.5T + 67T^{2} \)
71 \( 1 + (-1.86 - 1.35i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.05 - 12.4i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (3.44 - 2.50i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.12 + 2.26i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + (-5.61 + 4.08i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.610054446572483312309701637541, −8.599667981108759583149206482796, −7.903582183908209344985804052557, −7.02939035112464096648656775989, −6.73364866034601097608658422977, −5.50936900309018497593591198959, −4.58501384649562579837914316883, −2.61792871808489284388365221305, −1.37210258007173335002213708066, −0.092100147075648966464632166583, 2.28610592867644373292173517909, 3.83398165105683257220860027557, 4.08520558743107833567343166295, 5.59516723527157445150507682331, 6.23389912694560503850542359353, 7.73906063758418252797816086070, 8.712579197008365574810925885722, 9.146041476810414498106918359417, 10.30208305722464990825809909633, 10.76220947225421109953981939425

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