Properties

Label 2-175-175.17-c3-0-28
Degree $2$
Conductor $175$
Sign $0.0103 + 0.999i$
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  + (1.13 − 0.0597i)2-s + (−4.65 + 5.74i)3-s + (−6.66 + 0.700i)4-s + (−0.689 + 11.1i)5-s + (−4.95 + 6.81i)6-s + (−17.8 − 4.85i)7-s + (−16.5 + 2.62i)8-s + (−5.73 − 26.9i)9-s + (−0.118 + 12.7i)10-s + (9.78 + 2.07i)11-s + (26.9 − 41.5i)12-s + (62.8 − 32.0i)13-s + (−20.6 − 4.46i)14-s + (−60.8 − 55.8i)15-s + (33.7 − 7.16i)16-s + (9.43 − 24.5i)17-s + ⋯
L(s)  = 1  + (0.402 − 0.0211i)2-s + (−0.894 + 1.10i)3-s + (−0.832 + 0.0875i)4-s + (−0.0616 + 0.998i)5-s + (−0.337 + 0.464i)6-s + (−0.965 − 0.262i)7-s + (−0.731 + 0.115i)8-s + (−0.212 − 0.999i)9-s + (−0.00375 + 0.403i)10-s + (0.268 + 0.0569i)11-s + (0.648 − 0.998i)12-s + (1.34 − 0.682i)13-s + (−0.394 − 0.0852i)14-s + (−1.04 − 0.961i)15-s + (0.526 − 0.111i)16-s + (0.134 − 0.350i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0103 + 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0103 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.0103 + 0.999i$
Analytic conductor: \(10.3253\)
Root analytic conductor: \(3.21330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :3/2),\ 0.0103 + 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0148279 - 0.0146747i\)
\(L(\frac12)\) \(\approx\) \(0.0148279 - 0.0146747i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.689 - 11.1i)T \)
7 \( 1 + (17.8 + 4.85i)T \)
good2 \( 1 + (-1.13 + 0.0597i)T + (7.95 - 0.836i)T^{2} \)
3 \( 1 + (4.65 - 5.74i)T + (-5.61 - 26.4i)T^{2} \)
11 \( 1 + (-9.78 - 2.07i)T + (1.21e3 + 541. i)T^{2} \)
13 \( 1 + (-62.8 + 32.0i)T + (1.29e3 - 1.77e3i)T^{2} \)
17 \( 1 + (-9.43 + 24.5i)T + (-3.65e3 - 3.28e3i)T^{2} \)
19 \( 1 + (9.26 - 88.1i)T + (-6.70e3 - 1.42e3i)T^{2} \)
23 \( 1 + (-6.88 - 131. i)T + (-1.21e4 + 1.27e3i)T^{2} \)
29 \( 1 + (182. + 250. i)T + (-7.53e3 + 2.31e4i)T^{2} \)
31 \( 1 + (97.0 + 218. i)T + (-1.99e4 + 2.21e4i)T^{2} \)
37 \( 1 + (-20.9 - 13.6i)T + (2.06e4 + 4.62e4i)T^{2} \)
41 \( 1 + (201. + 65.6i)T + (5.57e4 + 4.05e4i)T^{2} \)
43 \( 1 + (370. - 370. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-496. + 190. i)T + (7.71e4 - 6.94e4i)T^{2} \)
53 \( 1 + (63.1 + 51.1i)T + (3.09e4 + 1.45e5i)T^{2} \)
59 \( 1 + (169. + 188. i)T + (-2.14e4 + 2.04e5i)T^{2} \)
61 \( 1 + (388. + 349. i)T + (2.37e4 + 2.25e5i)T^{2} \)
67 \( 1 + (37.4 + 14.3i)T + (2.23e5 + 2.01e5i)T^{2} \)
71 \( 1 + (-161. + 117. i)T + (1.10e5 - 3.40e5i)T^{2} \)
73 \( 1 + (-406. - 625. i)T + (-1.58e5 + 3.55e5i)T^{2} \)
79 \( 1 + (-23.1 + 51.9i)T + (-3.29e5 - 3.66e5i)T^{2} \)
83 \( 1 + (92.2 + 582. i)T + (-5.43e5 + 1.76e5i)T^{2} \)
89 \( 1 + (247. - 275. i)T + (-7.36e4 - 7.01e5i)T^{2} \)
97 \( 1 + (18.5 - 116. i)T + (-8.68e5 - 2.82e5i)T^{2} \)
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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

−11.81005958850736730329337193322, −10.99259068807458548240129975249, −9.968476093295875716294243555580, −9.496600985137906484304546830716, −7.84849818156488234339233940166, −6.15789831120040066765240674881, −5.62386133622373689622189843368, −3.94673058613932491008978915841, −3.50126331129900013691467979910, −0.01071997214894981712101205303, 1.27545021796703221015024229010, 3.72433515379229636449263162692, 5.12316827637382785219259340653, 6.06015829332649472268522066161, 6.92109699855181882202049364916, 8.703922048881203356040253558513, 9.093973573079360476171658190053, 10.74179191052972269358683855103, 12.01104147877592182430295299428, 12.60795802208276155530759942668

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