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Properties

Label 2-770-385.164-c1-0-32
Degree $2$
Conductor $770$
Sign $-0.0853 + 0.996i$
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 + (−1.23 + 2.14i)3-s + (−0.499 + 0.866i)4-s + (2.22 − 0.192i)5-s + 2.47·6-s + (−2.62 + 0.300i)7-s + 0.999·8-s + (−1.57 − 2.72i)9-s + (−1.28 − 1.83i)10-s + (−1.96 − 2.66i)11-s + (−1.23 − 2.14i)12-s − 3.09i·13-s + (1.57 + 2.12i)14-s + (−2.34 + 5.02i)15-s + (−0.5 − 0.866i)16-s + (−3.53 − 2.03i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.715 + 1.23i)3-s + (−0.249 + 0.433i)4-s + (0.996 − 0.0859i)5-s + 1.01·6-s + (−0.993 + 0.113i)7-s + 0.353·8-s + (−0.524 − 0.908i)9-s + (−0.404 − 0.579i)10-s + (−0.593 − 0.804i)11-s + (−0.357 − 0.619i)12-s − 0.858i·13-s + (0.420 + 0.568i)14-s + (−0.606 + 1.29i)15-s + (−0.125 − 0.216i)16-s + (−0.856 − 0.494i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0853 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0853 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.0853 + 0.996i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.0853 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.379914 - 0.413856i\)
\(L(\frac12)\) \(\approx\) \(0.379914 - 0.413856i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.22 + 0.192i)T \)
7 \( 1 + (2.62 - 0.300i)T \)
11 \( 1 + (1.96 + 2.66i)T \)
good3 \( 1 + (1.23 - 2.14i)T + (-1.5 - 2.59i)T^{2} \)
13 \( 1 + 3.09iT - 13T^{2} \)
17 \( 1 + (3.53 + 2.03i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.553 - 0.959i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.755 + 0.435i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.09iT - 29T^{2} \)
31 \( 1 + (5.80 + 3.35i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.68 + 3.85i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.60T + 41T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 + (0.969 + 1.67i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.95 + 5.74i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.54 + 1.46i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.51 + 2.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.7 + 7.38i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.20T + 71T^{2} \)
73 \( 1 + (4.91 + 2.83i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.86 - 1.65i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.69iT - 83T^{2} \)
89 \( 1 + (5.61 - 3.24i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.754T + 97T^{2} \)
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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

−10.13799462932428437397354968669, −9.436505582918576061042062382194, −8.982072876333719004383920687458, −7.63056611434652322154796956014, −6.16134672911175449058699824478, −5.63986454775538983417850167878, −4.64794903333496058548582565927, −3.45670556551752663559708946642, −2.50265516704376668642532619552, −0.35283187616878462540466783495, 1.40922513802763751894292820863, 2.52885147332138221287479709995, 4.50281098155723786967089105996, 5.72832820857854549152705935777, 6.29049440641881325718049590859, 6.98167471710918189933689496234, 7.53890654080929110764412142388, 8.955063768375478979553906187549, 9.533182355590052321581504103256, 10.48045513564006166451383567348

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