Properties

Label 2-175-5.2-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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−5.12 − 5.12i)2-s + (−10.3 + 10.3i)3-s + 36.5i·4-s + 106.·6-s + (−13.0 − 13.0i)7-s + (105. − 105. i)8-s − 132. i·9-s − 135.·11-s + (−378. − 378. i)12-s + (35.4 − 35.4i)13-s + 134. i·14-s − 497.·16-s + (−146. − 146. i)17-s + (−680. + 680. i)18-s − 408. i·19-s + ⋯
L(s)  = 1  + (−1.28 − 1.28i)2-s + (−1.14 + 1.14i)3-s + 2.28i·4-s + 2.94·6-s + (−0.267 − 0.267i)7-s + (1.64 − 1.64i)8-s − 1.63i·9-s − 1.12·11-s + (−2.62 − 2.62i)12-s + (0.209 − 0.209i)13-s + 0.685i·14-s − 1.94·16-s + (−0.506 − 0.506i)17-s + (−2.10 + 2.10i)18-s − 1.13i·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: $\chi_{175} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :2),\ 0.973 - 0.229i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.2860051812\)
\(L(\frac12)\) \(\approx\) \(0.2860051812\)
\(L(3)\) 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 \( 1 + (5.12 + 5.12i)T + 16iT^{2} \)
3 \( 1 + (10.3 - 10.3i)T - 81iT^{2} \)
11 \( 1 + 135.T + 1.46e4T^{2} \)
13 \( 1 + (-35.4 + 35.4i)T - 2.85e4iT^{2} \)
17 \( 1 + (146. + 146. i)T + 8.35e4iT^{2} \)
19 \( 1 + 408. iT - 1.30e5T^{2} \)
23 \( 1 + (212. - 212. i)T - 2.79e5iT^{2} \)
29 \( 1 - 195. iT - 7.07e5T^{2} \)
31 \( 1 - 965.T + 9.23e5T^{2} \)
37 \( 1 + (925. + 925. i)T + 1.87e6iT^{2} \)
41 \( 1 + 2.29e3T + 2.82e6T^{2} \)
43 \( 1 + (-1.19e3 + 1.19e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (-1.55e3 - 1.55e3i)T + 4.87e6iT^{2} \)
53 \( 1 + (645. - 645. i)T - 7.89e6iT^{2} \)
59 \( 1 - 2.29e3iT - 1.21e7T^{2} \)
61 \( 1 - 3.19e3T + 1.38e7T^{2} \)
67 \( 1 + (-6.11e3 - 6.11e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 740.T + 2.54e7T^{2} \)
73 \( 1 + (1.76e3 - 1.76e3i)T - 2.83e7iT^{2} \)
79 \( 1 - 8.40e3iT - 3.89e7T^{2} \)
83 \( 1 + (5.89e3 - 5.89e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 4.76e3iT - 6.27e7T^{2} \)
97 \( 1 + (687. + 687. i)T + 8.85e7iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.52272395501365503065352953584, −10.88710473867831711832449736634, −10.24055589837021082505566213197, −9.530220535123677563183788288660, −8.473998768537776781256882945751, −7.06458201760058168819511782808, −5.34270949518639009863193133118, −4.06712453495741272742248544423, −2.70543492156349084780317969471, −0.60816961254894766566468217977, 0.35951293856581048817887109148, 1.79461738554919540284805983793, 5.18233503787510218057695002952, 6.10551660736810863710365327891, 6.71395721569180632564138553660, 7.81045659243645554190718638359, 8.462919137712394896541382158386, 9.983796849351963734688241009591, 10.72028428392456752505325818929, 11.91994874226571365931741962098

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