Properties

Label 2-770-385.142-c1-0-1
Degree $2$
Conductor $770$
Sign $-0.155 - 0.987i$
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  + (−0.258 − 0.965i)2-s + (−2.93 − 0.785i)3-s + (−0.866 + 0.499i)4-s + (0.860 + 2.06i)5-s + 3.03i·6-s + (−0.604 − 2.57i)7-s + (0.707 + 0.707i)8-s + (5.37 + 3.10i)9-s + (1.77 − 1.36i)10-s + (0.981 − 3.16i)11-s + (2.93 − 0.785i)12-s + (−1.57 − 1.57i)13-s + (−2.33 + 1.25i)14-s + (−0.899 − 6.72i)15-s + (0.500 − 0.866i)16-s + (−5.75 − 1.54i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−1.69 − 0.453i)3-s + (−0.433 + 0.249i)4-s + (0.384 + 0.923i)5-s + 1.23i·6-s + (−0.228 − 0.973i)7-s + (0.249 + 0.249i)8-s + (1.79 + 1.03i)9-s + (0.560 − 0.431i)10-s + (0.296 − 0.955i)11-s + (0.846 − 0.226i)12-s + (−0.435 − 0.435i)13-s + (−0.623 + 0.334i)14-s + (−0.232 − 1.73i)15-s + (0.125 − 0.216i)16-s + (−1.39 − 0.373i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.155 - 0.987i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.155 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0315221 + 0.0368879i\)
\(L(\frac12)\) \(\approx\) \(0.0315221 + 0.0368879i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (-0.860 - 2.06i)T \)
7 \( 1 + (0.604 + 2.57i)T \)
11 \( 1 + (-0.981 + 3.16i)T \)
good3 \( 1 + (2.93 + 0.785i)T + (2.59 + 1.5i)T^{2} \)
13 \( 1 + (1.57 + 1.57i)T + 13iT^{2} \)
17 \( 1 + (5.75 + 1.54i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.0868 + 0.150i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.910 - 3.39i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 6.62T + 29T^{2} \)
31 \( 1 + (1.64 + 2.84i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (11.1 - 3.00i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 4.78iT - 41T^{2} \)
43 \( 1 + (-3.69 - 3.69i)T + 43iT^{2} \)
47 \( 1 + (-1.43 + 0.383i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.50 - 1.74i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (12.2 - 7.07i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.80 + 1.62i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.436 + 1.62i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 + (2.05 - 7.67i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.40 - 7.62i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.26 + 2.26i)T + 83iT^{2} \)
89 \( 1 + (-0.191 - 0.110i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.97 - 7.97i)T + 97iT^{2} \)
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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

−10.55794818956290434786480999983, −10.30966190527688353912921949923, −9.150126523076176107021987713549, −7.69113208078888100332543870138, −6.87521567425208783010586215050, −6.28687371965414233549106826615, −5.27237332303780443656774568433, −4.19743382772727651728607554132, −2.88957649247666210761946828175, −1.29570965345068275709975481835, 0.03507561056260200293366692541, 1.85182268399391993191136784777, 4.34315055266502975118121045872, 4.84649410521690328388164330869, 5.63344084266803078502482790452, 6.43229048768399530136413419400, 7.06714810273663813509833900630, 8.705769632885835806200988452945, 9.147615238396788969177164459685, 10.08601676447673375195290834079

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