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Properties

Label 2-770-385.54-c1-0-37
Degree $2$
Conductor $770$
Sign $0.336 + 0.941i$
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  + (−0.5 + 0.866i)2-s + (−0.340 − 0.589i)3-s + (−0.499 − 0.866i)4-s + (1.37 − 1.76i)5-s + 0.681·6-s + (2.52 − 0.782i)7-s + 0.999·8-s + (1.26 − 2.19i)9-s + (0.839 + 2.07i)10-s + (−3.24 − 0.708i)11-s + (−0.340 + 0.589i)12-s − 2.05i·13-s + (−0.585 + 2.58i)14-s + (−1.50 − 0.210i)15-s + (−0.5 + 0.866i)16-s + (2.89 − 1.67i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.196 − 0.340i)3-s + (−0.249 − 0.433i)4-s + (0.614 − 0.788i)5-s + 0.278·6-s + (0.955 − 0.295i)7-s + 0.353·8-s + (0.422 − 0.732i)9-s + (0.265 + 0.655i)10-s + (−0.976 − 0.213i)11-s + (−0.0982 + 0.170i)12-s − 0.568i·13-s + (−0.156 + 0.689i)14-s + (−0.389 − 0.0543i)15-s + (−0.125 + 0.216i)16-s + (0.701 − 0.405i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.336 + 0.941i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.336 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04359 - 0.735065i\)
\(L(\frac12)\) \(\approx\) \(1.04359 - 0.735065i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-1.37 + 1.76i)T \)
7 \( 1 + (-2.52 + 0.782i)T \)
11 \( 1 + (3.24 + 0.708i)T \)
good3 \( 1 + (0.340 + 0.589i)T + (-1.5 + 2.59i)T^{2} \)
13 \( 1 + 2.05iT - 13T^{2} \)
17 \( 1 + (-2.89 + 1.67i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.31 - 4.00i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.24 + 1.29i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.208iT - 29T^{2} \)
31 \( 1 + (7.69 - 4.44i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.14 - 1.81i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.23T + 41T^{2} \)
43 \( 1 - 6.55T + 43T^{2} \)
47 \( 1 + (-6.57 + 11.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.20 - 1.84i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.99 + 2.88i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.38 + 9.32i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.44 - 0.836i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.74T + 71T^{2} \)
73 \( 1 + (6.62 - 3.82i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-13.3 - 7.71i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.84iT - 83T^{2} \)
89 \( 1 + (-15.0 - 8.67i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.95T + 97T^{2} \)
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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

−10.12794555843803270780626859359, −9.195544274034314258176342177483, −8.244485317003906116240744022467, −7.75018660939858962985317470997, −6.69371401383485089373981017340, −5.59453533091276323302875853263, −5.16033932307033525467492738575, −3.86773430868326598635621199380, −1.94931578860110714954180281340, −0.76238324479436689116786175250, 1.83361389184751005823847673509, 2.54778828097305169254206874699, 4.05425814160460174091159522714, 5.04468197619501047857059010562, 5.89629531669528955482903711115, 7.34488117340759175988926781626, 7.83892256418208063725503293577, 9.018617678605286132761729491301, 9.809835748389364324104486664227, 10.71370513269941108815894922998

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