Properties

Label 2-570-95.37-c1-0-15
Degree $2$
Conductor $570$
Sign $-0.796 + 0.604i$
Analytic cond. $4.55147$
Root an. cond. $2.13341$
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 + (−0.707 − 0.707i)3-s − 1.00i·4-s + (2.23 − 0.0328i)5-s − 1.00·6-s + (−3.17 − 3.17i)7-s + (−0.707 − 0.707i)8-s + 1.00i·9-s + (1.55 − 1.60i)10-s + 1.49·11-s + (−0.707 + 0.707i)12-s + (−2.38 − 2.38i)13-s − 4.49·14-s + (−1.60 − 1.55i)15-s − 1.00·16-s + (0.853 + 0.853i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s − 0.500i·4-s + (0.999 − 0.0146i)5-s − 0.408·6-s + (−1.20 − 1.20i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (0.492 − 0.507i)10-s + 0.451·11-s + (−0.204 + 0.204i)12-s + (−0.662 − 0.662i)13-s − 1.20·14-s + (−0.414 − 0.402i)15-s − 0.250·16-s + (0.206 + 0.206i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 + 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.796 + 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(570\)    =    \(2 \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.796 + 0.604i$
Analytic conductor: \(4.55147\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{570} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 570,\ (\ :1/2),\ -0.796 + 0.604i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.475207 - 1.41368i\)
\(L(\frac12)\) \(\approx\) \(0.475207 - 1.41368i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-2.23 + 0.0328i)T \)
19 \( 1 + (4.34 - 0.349i)T \)
good7 \( 1 + (3.17 + 3.17i)T + 7iT^{2} \)
11 \( 1 - 1.49T + 11T^{2} \)
13 \( 1 + (2.38 + 2.38i)T + 13iT^{2} \)
17 \( 1 + (-0.853 - 0.853i)T + 17iT^{2} \)
23 \( 1 + (-4.67 + 4.67i)T - 23iT^{2} \)
29 \( 1 + 5.40T + 29T^{2} \)
31 \( 1 - 0.364iT - 31T^{2} \)
37 \( 1 + (-4.29 + 4.29i)T - 37iT^{2} \)
41 \( 1 - 1.79iT - 41T^{2} \)
43 \( 1 + (-0.623 + 0.623i)T - 43iT^{2} \)
47 \( 1 + (5.24 + 5.24i)T + 47iT^{2} \)
53 \( 1 + (-5.27 - 5.27i)T + 53iT^{2} \)
59 \( 1 + 1.49T + 59T^{2} \)
61 \( 1 - 15.2T + 61T^{2} \)
67 \( 1 + (-0.989 + 0.989i)T - 67iT^{2} \)
71 \( 1 + 8.98iT - 71T^{2} \)
73 \( 1 + (11.2 - 11.2i)T - 73iT^{2} \)
79 \( 1 - 7.95T + 79T^{2} \)
83 \( 1 + (-3.94 + 3.94i)T - 83iT^{2} \)
89 \( 1 - 9.30T + 89T^{2} \)
97 \( 1 + (-8.79 + 8.79i)T - 97iT^{2} \)
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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

−10.37288558051192516945924182065, −9.903201635334683238963634665676, −8.892882896798232083384448938945, −7.33908246765043916335554789300, −6.58109146596018730365592230570, −5.88127257012872593910510130964, −4.72870190697725718046133959991, −3.54122839910973134578611910743, −2.31292943802270084438683651376, −0.74186747329499734012610079342, 2.27439687531762654820476355818, 3.41888116718537505659935008509, 4.81765167370605802387502583986, 5.68714110888856510864354090632, 6.33661093175563584813136977704, 7.07668094714270647871556113630, 8.734050221389551973218583107204, 9.393464199977239074487854254027, 9.902121942449463189615616348883, 11.23602541868059741778838466788

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