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Properties

Label 2-770-55.32-c1-0-18
Degree $2$
Conductor $770$
Sign $0.907 + 0.420i$
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  + (0.707 − 0.707i)2-s + (−2.29 + 2.29i)3-s − 1.00i·4-s + (2.01 + 0.959i)5-s + 3.24i·6-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s − 7.56i·9-s + (2.10 − 0.749i)10-s + (1.71 − 2.83i)11-s + (2.29 + 2.29i)12-s + (−3.57 − 3.57i)13-s − 1.00i·14-s + (−6.84 + 2.43i)15-s − 1.00·16-s + (2.70 − 2.70i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−1.32 + 1.32i)3-s − 0.500i·4-s + (0.903 + 0.429i)5-s + 1.32i·6-s + (0.267 − 0.267i)7-s + (−0.250 − 0.250i)8-s − 2.52i·9-s + (0.666 − 0.237i)10-s + (0.517 − 0.855i)11-s + (0.663 + 0.663i)12-s + (−0.991 − 0.991i)13-s − 0.267i·14-s + (−1.76 + 0.628i)15-s − 0.250·16-s + (0.655 − 0.655i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 + 0.420i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.907 + 0.420i$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{770} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ 0.907 + 0.420i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47193 - 0.324230i\)
\(L(\frac12)\) \(\approx\) \(1.47193 - 0.324230i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-2.01 - 0.959i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + (-1.71 + 2.83i)T \)
good3 \( 1 + (2.29 - 2.29i)T - 3iT^{2} \)
13 \( 1 + (3.57 + 3.57i)T + 13iT^{2} \)
17 \( 1 + (-2.70 + 2.70i)T - 17iT^{2} \)
19 \( 1 - 6.45T + 19T^{2} \)
23 \( 1 + (3.60 - 3.60i)T - 23iT^{2} \)
29 \( 1 - 0.0794T + 29T^{2} \)
31 \( 1 - 7.10T + 31T^{2} \)
37 \( 1 + (6.54 + 6.54i)T + 37iT^{2} \)
41 \( 1 + 2.31iT - 41T^{2} \)
43 \( 1 + (-1.39 - 1.39i)T + 43iT^{2} \)
47 \( 1 + (-3.51 - 3.51i)T + 47iT^{2} \)
53 \( 1 + (-6.36 + 6.36i)T - 53iT^{2} \)
59 \( 1 - 2.39iT - 59T^{2} \)
61 \( 1 - 3.28iT - 61T^{2} \)
67 \( 1 + (4.69 + 4.69i)T + 67iT^{2} \)
71 \( 1 + 5.70T + 71T^{2} \)
73 \( 1 + (-5.28 - 5.28i)T + 73iT^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 + (-0.760 - 0.760i)T + 83iT^{2} \)
89 \( 1 + 6.82iT - 89T^{2} \)
97 \( 1 + (-8.02 - 8.02i)T + 97iT^{2} \)
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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

−10.27786210885609649264475138105, −9.858547681443051091917414134916, −9.156896691458498720327853989772, −7.41828467017930553165933102328, −6.25831651433878716265740274113, −5.41219640396842116219392778604, −5.17749044047809908202565883668, −3.82688231816518639660613266592, −2.97045774394028002666975194422, −0.880455583597338728460100392071, 1.36029945725580533171735212412, 2.32292448904239682656234202270, 4.57631197630610252966343358105, 5.20072044564519348051923056163, 6.03663013697554064753791041962, 6.71940358120269919843001283623, 7.40878849737704620425256285186, 8.355672729276791369884176178248, 9.615124565917881888741948842219, 10.42209379426773771165957297636

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