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Properties

Label 2-770-55.2-c1-0-9
Degree $2$
Conductor $770$
Sign $0.826 - 0.562i$
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  + (−0.453 − 0.891i)2-s + (−0.243 + 1.53i)3-s + (−0.587 + 0.809i)4-s + (−1.06 − 1.96i)5-s + (1.48 − 0.481i)6-s + (0.987 − 0.156i)7-s + (0.987 + 0.156i)8-s + (0.543 + 0.176i)9-s + (−1.27 + 1.83i)10-s + (−2.88 + 1.62i)11-s + (−1.10 − 1.10i)12-s + (−2.80 + 1.42i)13-s + (−0.587 − 0.809i)14-s + (3.28 − 1.15i)15-s + (−0.309 − 0.951i)16-s + (5.56 + 2.83i)17-s + ⋯
L(s)  = 1  + (−0.321 − 0.630i)2-s + (−0.140 + 0.888i)3-s + (−0.293 + 0.404i)4-s + (−0.474 − 0.880i)5-s + (0.605 − 0.196i)6-s + (0.373 − 0.0591i)7-s + (0.349 + 0.0553i)8-s + (0.181 + 0.0589i)9-s + (−0.402 + 0.581i)10-s + (−0.870 + 0.491i)11-s + (−0.318 − 0.318i)12-s + (−0.778 + 0.396i)13-s + (−0.157 − 0.216i)14-s + (0.848 − 0.297i)15-s + (−0.0772 − 0.237i)16-s + (1.34 + 0.687i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 - 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.826 - 0.562i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.826 - 0.562i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00473 + 0.309197i\)
\(L(\frac12)\) \(\approx\) \(1.00473 + 0.309197i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.453 + 0.891i)T \)
5 \( 1 + (1.06 + 1.96i)T \)
7 \( 1 + (-0.987 + 0.156i)T \)
11 \( 1 + (2.88 - 1.62i)T \)
good3 \( 1 + (0.243 - 1.53i)T + (-2.85 - 0.927i)T^{2} \)
13 \( 1 + (2.80 - 1.42i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (-5.56 - 2.83i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-3.02 + 2.19i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-3.25 + 3.25i)T - 23iT^{2} \)
29 \( 1 + (0.846 + 0.615i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.15 - 3.56i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.0602 + 0.380i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (-7.07 - 9.74i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (-4.13 - 4.13i)T + 43iT^{2} \)
47 \( 1 + (-10.8 - 1.71i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (-4.93 - 9.69i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (-2.40 + 3.30i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (14.7 - 4.77i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (-3.69 - 3.69i)T + 67iT^{2} \)
71 \( 1 + (1.40 + 4.31i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.06 - 13.0i)T + (-69.4 + 22.5i)T^{2} \)
79 \( 1 + (-3.07 + 9.45i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-7.34 + 14.4i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 - 8.22iT - 89T^{2} \)
97 \( 1 + (10.6 - 5.42i)T + (57.0 - 78.4i)T^{2} \)
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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

−10.41359752963478965816936839259, −9.583137963901160979116649413097, −9.021357116073344251394541815220, −7.84416511883733278006354421539, −7.40560832105448047433516791903, −5.50491533161340911708539128992, −4.74124170297588641321050878944, −4.15456419429179706517620742277, −2.83145786126999598680812449965, −1.23582504482869592272001210028, 0.71735233379407539340872159117, 2.42263399162898972944491893882, 3.67043170762119003291360433310, 5.24736200557075940488664119942, 5.88242867472931849174395784246, 7.21776329492606741011849446258, 7.45954940503448344729777408528, 8.031835674073225661765713081302, 9.378666850555578168133020570232, 10.22418237049992013331871560770

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