Properties

Label 2-175-5.3-c4-0-7
Degree $2$
Conductor $175$
Sign $-0.973 - 0.229i$
Analytic cond. $18.0897$
Root an. cond. $4.25320$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.01 + 5.01i)2-s + (5.80 + 5.80i)3-s − 34.2i·4-s − 58.1·6-s + (13.0 − 13.0i)7-s + (91.7 + 91.7i)8-s − 13.7i·9-s + 75.9·11-s + (198. − 198. i)12-s + (−87.6 − 87.6i)13-s + 131. i·14-s − 371.·16-s + (−230. + 230. i)17-s + (68.7 + 68.7i)18-s + 527. i·19-s + ⋯
L(s)  = 1  + (−1.25 + 1.25i)2-s + (0.644 + 0.644i)3-s − 2.14i·4-s − 1.61·6-s + (0.267 − 0.267i)7-s + (1.43 + 1.43i)8-s − 0.169i·9-s + 0.627·11-s + (1.38 − 1.38i)12-s + (−0.518 − 0.518i)13-s + 0.670i·14-s − 1.45·16-s + (−0.798 + 0.798i)17-s + (0.212 + 0.212i)18-s + 1.46i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.973 - 0.229i$
Analytic conductor: \(18.0897\)
Root analytic conductor: \(4.25320\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :2),\ -0.973 - 0.229i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.9976064526\)
\(L(\frac12)\) \(\approx\) \(0.9976064526\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + (-13.0 + 13.0i)T \)
good2 \( 1 + (5.01 - 5.01i)T - 16iT^{2} \)
3 \( 1 + (-5.80 - 5.80i)T + 81iT^{2} \)
11 \( 1 - 75.9T + 1.46e4T^{2} \)
13 \( 1 + (87.6 + 87.6i)T + 2.85e4iT^{2} \)
17 \( 1 + (230. - 230. i)T - 8.35e4iT^{2} \)
19 \( 1 - 527. iT - 1.30e5T^{2} \)
23 \( 1 + (-587. - 587. i)T + 2.79e5iT^{2} \)
29 \( 1 - 1.58e3iT - 7.07e5T^{2} \)
31 \( 1 + 588.T + 9.23e5T^{2} \)
37 \( 1 + (629. - 629. i)T - 1.87e6iT^{2} \)
41 \( 1 + 235.T + 2.82e6T^{2} \)
43 \( 1 + (-1.36e3 - 1.36e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (-1.62e3 + 1.62e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (845. + 845. i)T + 7.89e6iT^{2} \)
59 \( 1 + 1.60e3iT - 1.21e7T^{2} \)
61 \( 1 + 2.51e3T + 1.38e7T^{2} \)
67 \( 1 + (3.67e3 - 3.67e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 6.83e3T + 2.54e7T^{2} \)
73 \( 1 + (-1.60e3 - 1.60e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 3.82e3iT - 3.89e7T^{2} \)
83 \( 1 + (-4.12e3 - 4.12e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.14e3iT - 6.27e7T^{2} \)
97 \( 1 + (1.53e3 - 1.53e3i)T - 8.85e7iT^{2} \)
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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

−12.52791123811963677755725386924, −10.91800742198256484035656663415, −10.06360157333473326879859297535, −9.194384846032311531567897015746, −8.539688364575683824141300494972, −7.51624777421894443094999057647, −6.51324362456432382494514026718, −5.24366559330879101622399130969, −3.64766915546095577691687873950, −1.34322241374148616905033790468, 0.56356401752487865726108554376, 2.06778126769764552763564884064, 2.74489705789403773934846578712, 4.53948629235008023272501620428, 6.89758779270117335034231532524, 7.74395972310477091417671199646, 9.015682092437052274070206502950, 9.145830257029463103688317394895, 10.68192583186184255934465284833, 11.41149461276037324815565381947

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