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Properties

Label 2-770-385.349-c1-0-41
Degree $2$
Conductor $770$
Sign $-0.967 + 0.254i$
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 + (−0.934 − 2.87i)3-s + (0.309 − 0.951i)4-s + (2.04 − 0.902i)5-s + (−2.44 − 1.77i)6-s + (2.63 + 0.203i)7-s + (−0.309 − 0.951i)8-s + (−4.97 + 3.61i)9-s + (1.12 − 1.93i)10-s + (−2.96 − 1.48i)11-s − 3.02·12-s + (−0.321 − 0.442i)13-s + (2.25 − 1.38i)14-s + (−4.50 − 5.03i)15-s + (−0.809 − 0.587i)16-s + (3.59 − 4.94i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.539 − 1.66i)3-s + (0.154 − 0.475i)4-s + (0.914 − 0.403i)5-s + (−0.998 − 0.725i)6-s + (0.997 + 0.0768i)7-s + (−0.109 − 0.336i)8-s + (−1.65 + 1.20i)9-s + (0.355 − 0.611i)10-s + (−0.893 − 0.448i)11-s − 0.872·12-s + (−0.0892 − 0.122i)13-s + (0.602 − 0.370i)14-s + (−1.16 − 1.30i)15-s + (−0.202 − 0.146i)16-s + (0.871 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.967 + 0.254i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.967 + 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.263521 - 2.03799i\)
\(L(\frac12)\) \(\approx\) \(0.263521 - 2.03799i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.04 + 0.902i)T \)
7 \( 1 + (-2.63 - 0.203i)T \)
11 \( 1 + (2.96 + 1.48i)T \)
good3 \( 1 + (0.934 + 2.87i)T + (-2.42 + 1.76i)T^{2} \)
13 \( 1 + (0.321 + 0.442i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.59 + 4.94i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.0567 - 0.174i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 8.00iT - 23T^{2} \)
29 \( 1 + (4.80 + 1.55i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.63 + 3.63i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-4.53 - 1.47i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.44 - 10.5i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 2.20T + 43T^{2} \)
47 \( 1 + (-0.0177 - 0.0547i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-7.85 - 10.8i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (5.59 + 1.81i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.98 - 2.89i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 9.61iT - 67T^{2} \)
71 \( 1 + (-0.266 - 0.193i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (6.98 + 2.26i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.23 + 3.08i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-3.14 + 4.32i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 - 1.52iT - 89T^{2} \)
97 \( 1 + (-8.96 + 6.51i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.09925523669426204918215504510, −9.077971678239625049290703390947, −7.75667106361048640311384254628, −7.50489731935364317477045044747, −6.04272091396959416946282026989, −5.59396547596942781675499032557, −4.88161311987561313718130122075, −2.87452851955529091297966975557, −1.89480797126400022771564489385, −0.963203438464845674842871508778, 2.28002816794820732037816222849, 3.63837999091507347731559115930, 4.56346790912069668078415507673, 5.36575114385666977104892583634, 5.81442979390079345757569804399, 7.05096727270276612012697305119, 8.250909505305441264975405547633, 9.101132526581622764267549130420, 10.24650030250289380305180794060, 10.50310175714984365345960939602

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