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Properties

Label 2-175-7.2-c3-0-21
Degree $2$
Conductor $175$
Sign $0.997 - 0.0706i$
Analytic cond. $10.3253$
Root an. cond. $3.21330$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.20 + 3.82i)2-s + (1.62 + 2.80i)3-s + (−5.74 − 9.94i)4-s − 14.3·6-s + (−16.1 − 9.14i)7-s + 15.3·8-s + (8.24 − 14.2i)9-s + (0.0710 + 0.123i)11-s + (18.6 − 32.2i)12-s + 32.1·13-s + (70.4 − 41.3i)14-s + (11.9 − 20.7i)16-s + (−57.1 − 99.0i)17-s + (36.3 + 63.0i)18-s + (−21.6 + 37.4i)19-s + ⋯
L(s)  = 1  + (−0.780 + 1.35i)2-s + (0.312 + 0.540i)3-s + (−0.717 − 1.24i)4-s − 0.973·6-s + (−0.869 − 0.493i)7-s + 0.679·8-s + (0.305 − 0.528i)9-s + (0.00194 + 0.00337i)11-s + (0.447 − 0.775i)12-s + 0.685·13-s + (1.34 − 0.790i)14-s + (0.187 − 0.324i)16-s + (−0.815 − 1.41i)17-s + (0.476 + 0.825i)18-s + (−0.260 + 0.452i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0706i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.997 - 0.0706i$
Analytic conductor: \(10.3253\)
Root analytic conductor: \(3.21330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :3/2),\ 0.997 - 0.0706i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.746826 + 0.0264190i\)
\(L(\frac12)\) \(\approx\) \(0.746826 + 0.0264190i\)
\(L(\frac{5}{2})\) 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 \( 1 + (16.1 + 9.14i)T \)
good2 \( 1 + (2.20 - 3.82i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (-1.62 - 2.80i)T + (-13.5 + 23.3i)T^{2} \)
11 \( 1 + (-0.0710 - 0.123i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 32.1T + 2.19e3T^{2} \)
17 \( 1 + (57.1 + 99.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (21.6 - 37.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-77.2 + 133. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 40.1T + 2.43e4T^{2} \)
31 \( 1 + (37.7 + 65.3i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (200. - 346. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 95.4T + 6.89e4T^{2} \)
43 \( 1 - 340.T + 7.95e4T^{2} \)
47 \( 1 + (3.74 - 6.48i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (338. + 586. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (398. + 689. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-378. + 655. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (370. + 641. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 37.0T + 3.57e5T^{2} \)
73 \( 1 + (-40.4 - 70.0i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-158. + 274. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 945.T + 5.71e5T^{2} \)
89 \( 1 + (391. - 678. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 393.T + 9.12e5T^{2} \)
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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

−12.40829537592478229807864523193, −10.85747441423551795243915261607, −9.737897684390995486412421068998, −9.201739799802816013860575091865, −8.219331503408069841046718738193, −6.88231215486654320412216756869, −6.41686348523769278359931107525, −4.79282411161444895706586429762, −3.31415683198840291245710616870, −0.44381878714096277187893656264, 1.46230159442991157565174763748, 2.60273616426828695698773556412, 3.87502284639498457921377694002, 5.94100655226129177340568679603, 7.31475374520168267770387888868, 8.651923546823838100460404959365, 9.155430338939792693972064695472, 10.45079361965578099777883756069, 11.01653833421437204358340154113, 12.27583347986876047191124748397

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