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Properties

Label 2-175-5.3-c4-0-23
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  + (−3.68 + 3.68i)2-s + (5.41 + 5.41i)3-s − 11.2i·4-s − 39.9·6-s + (−13.0 + 13.0i)7-s + (−17.5 − 17.5i)8-s − 22.3i·9-s + 58.8·11-s + (60.8 − 60.8i)12-s + (−90.6 − 90.6i)13-s − 96.6i·14-s + 309.·16-s + (−18.7 + 18.7i)17-s + (82.4 + 82.4i)18-s − 466. i·19-s + ⋯
L(s)  = 1  + (−0.922 + 0.922i)2-s + (0.601 + 0.601i)3-s − 0.701i·4-s − 1.11·6-s + (−0.267 + 0.267i)7-s + (−0.274 − 0.274i)8-s − 0.275i·9-s + 0.486·11-s + (0.422 − 0.422i)12-s + (−0.536 − 0.536i)13-s − 0.493i·14-s + 1.20·16-s + (−0.0648 + 0.0648i)17-s + (0.254 + 0.254i)18-s − 1.29i·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

\(L(\frac{5}{2})\) \(\approx\) \(0.8019560125\)
\(L(\frac12)\) \(\approx\) \(0.8019560125\)
\(L(3)\) 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 \( 1 + (3.68 - 3.68i)T - 16iT^{2} \)
3 \( 1 + (-5.41 - 5.41i)T + 81iT^{2} \)
11 \( 1 - 58.8T + 1.46e4T^{2} \)
13 \( 1 + (90.6 + 90.6i)T + 2.85e4iT^{2} \)
17 \( 1 + (18.7 - 18.7i)T - 8.35e4iT^{2} \)
19 \( 1 + 466. iT - 1.30e5T^{2} \)
23 \( 1 + (617. + 617. i)T + 2.79e5iT^{2} \)
29 \( 1 - 33.4iT - 7.07e5T^{2} \)
31 \( 1 + 1.02e3T + 9.23e5T^{2} \)
37 \( 1 + (-495. + 495. i)T - 1.87e6iT^{2} \)
41 \( 1 - 1.43e3T + 2.82e6T^{2} \)
43 \( 1 + (-2.15e3 - 2.15e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (-722. + 722. i)T - 4.87e6iT^{2} \)
53 \( 1 + (2.10e3 + 2.10e3i)T + 7.89e6iT^{2} \)
59 \( 1 - 3.85e3iT - 1.21e7T^{2} \)
61 \( 1 - 5.49e3T + 1.38e7T^{2} \)
67 \( 1 + (1.18e3 - 1.18e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 2.60e3T + 2.54e7T^{2} \)
73 \( 1 + (6.99e3 + 6.99e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 1.02e4iT - 3.89e7T^{2} \)
83 \( 1 + (-1.42e3 - 1.42e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 5.35e3iT - 6.27e7T^{2} \)
97 \( 1 + (-3.09e3 + 3.09e3i)T - 8.85e7iT^{2} \)
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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

−12.03570874512417350527260193225, −10.53952425940377562920810892045, −9.494060064499509898084829884921, −9.030609062480790587814997010006, −8.044811379451596821434685945825, −6.93648389923431718863048760380, −5.91886066899265081622042379913, −4.21103564319654268277377842414, −2.81985616659316682671141883172, −0.38767427210919069059064884232, 1.39115262442286114191830481126, 2.34166218753760869392771410646, 3.79652904137197980921306116549, 5.75812582189895119735275285458, 7.30479374903923726190116350819, 8.128588219973776962813978032828, 9.214864238772538330005669803594, 9.948614685066085464491475265158, 10.98429371903853469795093244305, 11.97127867122436170441290151809

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