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Properties

Label 2-770-385.54-c1-0-44
Degree $2$
Conductor $770$
Sign $-0.537 - 0.843i$
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 + (−1.44 − 2.49i)3-s + (−0.499 − 0.866i)4-s + (−2.16 − 0.568i)5-s + 2.88·6-s + (−0.391 − 2.61i)7-s + 0.999·8-s + (−2.64 + 4.58i)9-s + (1.57 − 1.58i)10-s + (−3.21 − 0.802i)11-s + (−1.44 + 2.49i)12-s − 6.76i·13-s + (2.46 + 0.969i)14-s + (1.69 + 6.21i)15-s + (−0.5 + 0.866i)16-s + (1.33 − 0.772i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.831 − 1.44i)3-s + (−0.249 − 0.433i)4-s + (−0.967 − 0.254i)5-s + 1.17·6-s + (−0.147 − 0.989i)7-s + 0.353·8-s + (−0.882 + 1.52i)9-s + (0.497 − 0.502i)10-s + (−0.970 − 0.241i)11-s + (−0.415 + 0.720i)12-s − 1.87i·13-s + (0.657 + 0.259i)14-s + (0.437 + 1.60i)15-s + (−0.125 + 0.216i)16-s + (0.324 − 0.187i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.537 - 0.843i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -0.537 - 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.120055 + 0.218865i\)
\(L(\frac12)\) \(\approx\) \(0.120055 + 0.218865i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.16 + 0.568i)T \)
7 \( 1 + (0.391 + 2.61i)T \)
11 \( 1 + (3.21 + 0.802i)T \)
good3 \( 1 + (1.44 + 2.49i)T + (-1.5 + 2.59i)T^{2} \)
13 \( 1 + 6.76iT - 13T^{2} \)
17 \( 1 + (-1.33 + 0.772i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.445 + 0.771i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.17 + 3.56i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.79iT - 29T^{2} \)
31 \( 1 + (-0.849 + 0.490i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.14 - 4.70i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.03T + 41T^{2} \)
43 \( 1 - 0.302T + 43T^{2} \)
47 \( 1 + (5.24 - 9.08i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.89 - 2.82i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.61 + 2.08i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.226 + 0.393i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.39 + 4.84i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.72T + 71T^{2} \)
73 \( 1 + (-2.80 + 1.61i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.09 + 4.09i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.01iT - 83T^{2} \)
89 \( 1 + (-6.83 - 3.94i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 19.1T + 97T^{2} \)
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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

−9.903632111072221735190905103190, −8.171628268461821262748085021702, −7.906441816533838080923290779115, −7.38130006598618500790222019111, −6.35704595025255866147176540619, −5.57741754921045775480450199921, −4.56995201216079318821859730676, −2.97164370517720997617365116244, −0.970305710801700471843787166654, −0.20425911426000767094427964089, 2.37248092193745445108459389598, 3.72926458206434277985705024655, 4.33475815543908729122554593221, 5.32094654020033103517158824388, 6.32502656224023825827060824573, 7.65077027494372085683210681434, 8.603180001222398355499598800127, 9.534731937904532911780054156714, 9.966196427328054245470586794573, 10.98260954538544014040175100506

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