Properties

Label 2-770-11.3-c1-0-15
Degree $2$
Conductor $770$
Sign $0.444 + 0.895i$
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 + (0.167 + 0.516i)3-s + (0.309 − 0.951i)4-s + (−0.809 − 0.587i)5-s + (0.439 + 0.319i)6-s + (−0.309 + 0.951i)7-s + (−0.309 − 0.951i)8-s + (2.18 − 1.58i)9-s − 10-s + (2.31 − 2.37i)11-s + 0.543·12-s + (1.34 − 0.974i)13-s + (0.309 + 0.951i)14-s + (0.167 − 0.516i)15-s + (−0.809 − 0.587i)16-s + (−1.65 − 1.20i)17-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (0.0969 + 0.298i)3-s + (0.154 − 0.475i)4-s + (−0.361 − 0.262i)5-s + (0.179 + 0.130i)6-s + (−0.116 + 0.359i)7-s + (−0.109 − 0.336i)8-s + (0.729 − 0.529i)9-s − 0.316·10-s + (0.698 − 0.716i)11-s + 0.156·12-s + (0.372 − 0.270i)13-s + (0.0825 + 0.254i)14-s + (0.0433 − 0.133i)15-s + (−0.202 − 0.146i)16-s + (−0.401 − 0.291i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.444 + 0.895i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.444 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83609 - 1.13908i\)
\(L(\frac12)\) \(\approx\) \(1.83609 - 1.13908i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 + 0.587i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (-2.31 + 2.37i)T \)
good3 \( 1 + (-0.167 - 0.516i)T + (-2.42 + 1.76i)T^{2} \)
13 \( 1 + (-1.34 + 0.974i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.65 + 1.20i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.14 - 3.52i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 4.99T + 23T^{2} \)
29 \( 1 + (-2.20 + 6.78i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-8.23 + 5.98i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.82 + 8.68i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.55 - 4.79i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 5.78T + 43T^{2} \)
47 \( 1 + (1.41 + 4.34i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (7.18 - 5.22i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.35 - 13.3i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.67 - 1.94i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 3.18T + 67T^{2} \)
71 \( 1 + (-6.31 - 4.58i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.36 - 10.3i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (9.06 - 6.58i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.29 + 2.39i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 3.32T + 89T^{2} \)
97 \( 1 + (7.44 - 5.40i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.10023220416909089215070351630, −9.516049108729263720018648390094, −8.574912742768027879169156760964, −7.62871541879983294953961333916, −6.32277660793231216810307680725, −5.79533106216810654643527665840, −4.30240270759695017704047661291, −3.92180882440431221631672125190, −2.64799907127074565783919260276, −1.02663368518423598755848479190, 1.62563585517634784846211905588, 3.08017868378714923018559016841, 4.27594681663631528646732382334, 4.82350934833993323975478282873, 6.43061207563747171617548222400, 6.82312527725551192073038010477, 7.70281141115535656915303570908, 8.518518237974080964238371680091, 9.643990752491384384735176442685, 10.53904474291669885166627943441

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