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Properties

Label 2-770-11.3-c1-0-3
Degree $2$
Conductor $770$
Sign $0.0246 - 0.999i$
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 + (0.361 + 1.11i)3-s + (0.309 − 0.951i)4-s + (−0.809 − 0.587i)5-s + (−0.947 − 0.688i)6-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (1.31 − 0.956i)9-s + 10-s + (−3.15 + 1.02i)11-s + 1.17·12-s + (−2.19 + 1.59i)13-s + (0.309 + 0.951i)14-s + (0.361 − 1.11i)15-s + (−0.809 − 0.587i)16-s + (4.32 + 3.14i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.208 + 0.643i)3-s + (0.154 − 0.475i)4-s + (−0.361 − 0.262i)5-s + (−0.386 − 0.281i)6-s + (0.116 − 0.359i)7-s + (0.109 + 0.336i)8-s + (0.438 − 0.318i)9-s + 0.316·10-s + (−0.951 + 0.308i)11-s + 0.338·12-s + (−0.609 + 0.442i)13-s + (0.0825 + 0.254i)14-s + (0.0934 − 0.287i)15-s + (−0.202 − 0.146i)16-s + (1.04 + 0.761i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0246 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0246 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.0246 - 0.999i$
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: \((2,\ 770,\ (\ :1/2),\ 0.0246 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.791530 + 0.772231i\)
\(L(\frac12)\) \(\approx\) \(0.791530 + 0.772231i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.15 - 1.02i)T \)
good3 \( 1 + (-0.361 - 1.11i)T + (-2.42 + 1.76i)T^{2} \)
13 \( 1 + (2.19 - 1.59i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.32 - 3.14i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.22 - 3.78i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 5.49T + 23T^{2} \)
29 \( 1 + (1.54 - 4.75i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.32 + 1.68i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.0125 + 0.0385i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.58 - 4.87i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 12.2T + 43T^{2} \)
47 \( 1 + (-3.05 - 9.41i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.68 - 2.67i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.180 - 0.556i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.87 + 4.26i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 14.4T + 67T^{2} \)
71 \( 1 + (6.64 + 4.82i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.50 + 13.8i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.46 - 3.96i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-12.6 - 9.15i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 2.73T + 89T^{2} \)
97 \( 1 + (-1.07 + 0.780i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.40343672259948340660211805339, −9.590729083380565205517227693647, −8.981897203109293179749758279704, −7.73218972063392278875870851518, −7.50434742296780965280609221150, −6.19333199179498137159439553831, −5.07150725510958823617571675352, −4.27593222279663805073270573898, −3.07456264673264895183269653848, −1.30341256162744403269131759346, 0.75643666298389778636423521083, 2.40349486086119960824061949296, 3.05623057799027715855863817503, 4.65688489794623829101061017285, 5.65900299765857466550038953934, 7.20116788938384723861108151113, 7.45206898897656156305557650563, 8.307583733362007115489423210663, 9.254076857395620030452191975275, 10.19178822431862183883843830321

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