↑

Properties

Label 2-175-175.17-c1-0-8
Degree $2$
Conductor $175$
Sign $0.862 + 0.505i$
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  + (−0.855 + 0.0448i)2-s + (−0.835 + 1.03i)3-s + (−1.25 + 0.132i)4-s + (1.02 − 1.98i)5-s + (0.668 − 0.920i)6-s + (−0.0574 − 2.64i)7-s + (2.76 − 0.437i)8-s + (0.256 + 1.20i)9-s + (−0.786 + 1.74i)10-s + (0.886 + 0.188i)11-s + (0.915 − 1.41i)12-s + (5.68 − 2.89i)13-s + (0.167 + 2.26i)14-s + (1.19 + 2.71i)15-s + (0.131 − 0.0279i)16-s + (0.276 − 0.720i)17-s + ⋯
L(s)  = 1  + (−0.605 + 0.0317i)2-s + (−0.482 + 0.595i)3-s + (−0.629 + 0.0661i)4-s + (0.457 − 0.889i)5-s + (0.273 − 0.375i)6-s + (−0.0217 − 0.999i)7-s + (0.977 − 0.154i)8-s + (0.0856 + 0.402i)9-s + (−0.248 + 0.552i)10-s + (0.267 + 0.0568i)11-s + (0.264 − 0.407i)12-s + (1.57 − 0.803i)13-s + (0.0448 + 0.604i)14-s + (0.309 + 0.701i)15-s + (0.0328 − 0.00698i)16-s + (0.0671 − 0.174i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 + 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.862 + 0.505i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ 0.862 + 0.505i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.682191 - 0.184995i\)
\(L(\frac12)\) \(\approx\) \(0.682191 - 0.184995i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.02 + 1.98i)T \)
7 \( 1 + (0.0574 + 2.64i)T \)
good2 \( 1 + (0.855 - 0.0448i)T + (1.98 - 0.209i)T^{2} \)
3 \( 1 + (0.835 - 1.03i)T + (-0.623 - 2.93i)T^{2} \)
11 \( 1 + (-0.886 - 0.188i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-5.68 + 2.89i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (-0.276 + 0.720i)T + (-12.6 - 11.3i)T^{2} \)
19 \( 1 + (0.0762 - 0.725i)T + (-18.5 - 3.95i)T^{2} \)
23 \( 1 + (0.328 + 6.25i)T + (-22.8 + 2.40i)T^{2} \)
29 \( 1 + (3.24 + 4.46i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.93 - 4.34i)T + (-20.7 + 23.0i)T^{2} \)
37 \( 1 + (-8.05 - 5.23i)T + (15.0 + 33.8i)T^{2} \)
41 \( 1 + (-5.14 - 1.67i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (3.27 - 3.27i)T - 43iT^{2} \)
47 \( 1 + (-0.361 + 0.138i)T + (34.9 - 31.4i)T^{2} \)
53 \( 1 + (6.15 + 4.98i)T + (11.0 + 51.8i)T^{2} \)
59 \( 1 + (0.380 + 0.422i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (-7.85 - 7.07i)T + (6.37 + 60.6i)T^{2} \)
67 \( 1 + (2.29 + 0.879i)T + (49.7 + 44.8i)T^{2} \)
71 \( 1 + (9.01 - 6.54i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-6.64 - 10.2i)T + (-29.6 + 66.6i)T^{2} \)
79 \( 1 + (-0.729 + 1.63i)T + (-52.8 - 58.7i)T^{2} \)
83 \( 1 + (0.805 + 5.08i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (-2.15 + 2.38i)T + (-9.30 - 88.5i)T^{2} \)
97 \( 1 + (-1.57 + 9.95i)T + (-92.2 - 29.9i)T^{2} \)
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

−12.99143573156106614197260558710, −11.29898391333352342732180572374, −10.35944461690107611119240557810, −9.801637514091566907673036636788, −8.592032961735806861738795616262, −7.86969780047023756815930850480, −6.12063621472703744175545749736, −4.84159331523015928646416953168, −4.03209276961467473599327134883, −1.01532523295867503228411784738, 1.64032126183438281381920837351, 3.72674549329707876346902004001, 5.67353943700095944025558367807, 6.39970906962381934679504879140, 7.66025075459226200557310877081, 9.033476496813826371687900459408, 9.503647333773493249050861031918, 10.94771502271461160223110663643, 11.57614559842464426416082359412, 12.86929668981004339993770949475

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