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Properties

Label 2-770-385.54-c1-0-3
Degree $2$
Conductor $770$
Sign $0.861 - 0.507i$
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 + (0.914 − 3.18i)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 + (0.275 − 0.961i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.861 - 0.507i$
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: \((2,\ 770,\ (\ :1/2),\ 0.861 - 0.507i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.333532 + 0.0909591i\)
\(L(\frac12)\) \(\approx\) \(0.333532 + 0.0909591i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.914 + 3.18i)T \)
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

−11.03747008175546210704482031169, −9.483332989067083421334112860065, −8.492290128803849840037075416437, −7.916403719341593526067965716643, −6.47335211297172356431284230725, −6.28529000493096073473232544600, −5.00158153038646953214033770350, −4.00863166455221727463946921942, −2.45736838512704549823187530546, −1.38939739933265204290902852531, 0.18236798635004754566260557097, 3.32940891967024858781560455579, 4.03222977874952584285942533074, 4.75869549727772074437184563084, 5.55309443739521868675714692374, 6.71378554484192952803280563357, 7.58865052911978090218013457609, 8.352028388361802667820719943538, 9.724169918759771092355460510046, 10.24462073785598274313516406195

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