↑

Properties

Label 2-175-7.4-c1-0-8
Degree $2$
Conductor $175$
Sign $0.0725 + 0.997i$
Analytic cond. $1.39738$
Root an. cond. $1.18210$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.207 − 0.358i)2-s + (1.20 − 2.09i)3-s + (0.914 − 1.58i)4-s − 6-s + (1.62 + 2.09i)7-s − 1.58·8-s + (−1.41 − 2.44i)9-s + (−2.41 + 4.18i)11-s + (−2.20 − 3.82i)12-s − 0.828·13-s + (0.414 − 1.01i)14-s + (−1.49 − 2.59i)16-s + (−0.414 + 0.717i)17-s + (−0.585 + 1.01i)18-s + (1.41 + 2.44i)19-s + ⋯
L(s)  = 1  + (−0.146 − 0.253i)2-s + (0.696 − 1.20i)3-s + (0.457 − 0.791i)4-s − 0.408·6-s + (0.612 + 0.790i)7-s − 0.560·8-s + (−0.471 − 0.816i)9-s + (−0.727 + 1.26i)11-s + (−0.637 − 1.10i)12-s − 0.229·13-s + (0.110 − 0.271i)14-s + (−0.374 − 0.649i)16-s + (−0.100 + 0.174i)17-s + (−0.138 + 0.239i)18-s + (0.324 + 0.561i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.0725 + 0.997i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ 0.0725 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04163 - 0.968608i\)
\(L(\frac12)\) \(\approx\) \(1.04163 - 0.968608i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 2.09i)T \)
good2 \( 1 + (0.207 + 0.358i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.20 + 2.09i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (2.41 - 4.18i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.828T + 13T^{2} \)
17 \( 1 + (0.414 - 0.717i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.41 - 2.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.20 + 2.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + (-3 + 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.17T + 41T^{2} \)
43 \( 1 + 6.41T + 43T^{2} \)
47 \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.41 - 5.91i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.24 + 10.8i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.74 - 9.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.20 + 10.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + (-2.41 + 4.18i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.58 + 7.94i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + (1.32 + 2.30i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.343T + 97T^{2} \)
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

−12.38639375290803436106153758483, −11.75768410863597629380427175936, −10.45214797264989957227944386906, −9.461877721834180559844678091312, −8.227603736651881293425160547535, −7.40473957804389206003968003085, −6.27453371851719097167305360405, −5.00133818761332780826369219697, −2.54355492331874252910419348469, −1.75796855752372390685593674987, 2.90723323392972382989691767426, 3.86847158550969055579115645997, 5.18494296427381168925440005700, 6.93150157492033961969785268208, 8.101552133121464899516032950856, 8.685819166400673554071254659739, 9.954239918959783901134030879030, 10.89434001304422595574616147211, 11.67848325561118867552669796989, 13.19807249139690787672266114180

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