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Properties

Label 2-770-11.3-c1-0-10
Degree $2$
Conductor $770$
Sign $0.773 - 0.634i$
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 + (0.357 + 1.10i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (−0.936 − 0.680i)6-s + (0.309 − 0.951i)7-s + (0.309 + 0.951i)8-s + (1.34 − 0.975i)9-s − 10-s + (2.83 + 1.72i)11-s + 1.15·12-s + (2.90 − 2.10i)13-s + (0.309 + 0.951i)14-s + (−0.357 + 1.10i)15-s + (−0.809 − 0.587i)16-s + (−3.47 − 2.52i)17-s + ⋯
L(s)  = 1  + (−0.572 + 0.415i)2-s + (0.206 + 0.635i)3-s + (0.154 − 0.475i)4-s + (0.361 + 0.262i)5-s + (−0.382 − 0.277i)6-s + (0.116 − 0.359i)7-s + (0.109 + 0.336i)8-s + (0.447 − 0.325i)9-s − 0.316·10-s + (0.854 + 0.519i)11-s + 0.334·12-s + (0.804 − 0.584i)13-s + (0.0825 + 0.254i)14-s + (−0.0923 + 0.284i)15-s + (−0.202 − 0.146i)16-s + (−0.841 − 0.611i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.773 - 0.634i$
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: \((2,\ 770,\ (\ :1/2),\ 0.773 - 0.634i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41357 + 0.505604i\)
\(L(\frac12)\) \(\approx\) \(1.41357 + 0.505604i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.83 - 1.72i)T \)
good3 \( 1 + (-0.357 - 1.10i)T + (-2.42 + 1.76i)T^{2} \)
13 \( 1 + (-2.90 + 2.10i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.47 + 2.52i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.51 + 4.65i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2.85T + 23T^{2} \)
29 \( 1 + (0.233 - 0.717i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-7.56 + 5.49i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.31 - 4.03i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.626 - 1.92i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 0.0689T + 43T^{2} \)
47 \( 1 + (-0.691 - 2.12i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (10.6 - 7.70i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.378 + 1.16i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.26 - 2.36i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 14.3T + 67T^{2} \)
71 \( 1 + (-4.69 - 3.40i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.80 - 8.62i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.07 + 2.95i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-13.4 - 9.77i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 9.92T + 89T^{2} \)
97 \( 1 + (-5.43 + 3.94i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.28614937235177836999713167401, −9.373929328025359851926843045227, −9.021309140018436976674920781874, −7.88339308419362746770477455245, −6.80605120308713236536938727897, −6.37564198098255345835199853995, −4.91988578304361038636963661899, −4.13699602207663247975811145938, −2.78825137544787951067980977247, −1.15264349750558412461025599871, 1.31467796133864270402299969782, 2.07210273618757422039036063534, 3.55252756838268209992250553313, 4.65206288483728661944923304310, 6.17375049988935413860503164397, 6.68572081592076488602818044860, 7.908025594018438283546928632096, 8.629268903988446235775562098583, 9.162322332192444381433622311020, 10.28014304823373863166092181526

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