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Properties

Label 2-175-175.3-c1-0-11
Degree $2$
Conductor $175$
Sign $0.994 + 0.100i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.423 + 0.651i)2-s + (0.402 − 1.04i)3-s + (0.567 + 1.27i)4-s + (0.567 − 2.16i)5-s + (0.513 + 0.706i)6-s + (1.24 − 2.33i)7-s + (−2.60 − 0.412i)8-s + (1.29 + 1.16i)9-s + (1.16 + 1.28i)10-s + (−0.888 − 0.986i)11-s + (1.56 − 0.0820i)12-s + (1.79 + 0.912i)13-s + (0.995 + 1.79i)14-s + (−2.03 − 1.46i)15-s + (−0.495 + 0.550i)16-s + (−0.585 + 0.722i)17-s + ⋯
L(s)  = 1  + (−0.299 + 0.460i)2-s + (0.232 − 0.605i)3-s + (0.283 + 0.637i)4-s + (0.253 − 0.967i)5-s + (0.209 + 0.288i)6-s + (0.470 − 0.882i)7-s + (−0.921 − 0.145i)8-s + (0.430 + 0.387i)9-s + (0.369 + 0.406i)10-s + (−0.267 − 0.297i)11-s + (0.451 − 0.0236i)12-s + (0.496 + 0.253i)13-s + (0.266 + 0.480i)14-s + (−0.526 − 0.378i)15-s + (−0.123 + 0.137i)16-s + (−0.141 + 0.175i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.994 + 0.100i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ 0.994 + 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21513 - 0.0614907i\)
\(L(\frac12)\) \(\approx\) \(1.21513 - 0.0614907i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.567 + 2.16i)T \)
7 \( 1 + (-1.24 + 2.33i)T \)
good2 \( 1 + (0.423 - 0.651i)T + (-0.813 - 1.82i)T^{2} \)
3 \( 1 + (-0.402 + 1.04i)T + (-2.22 - 2.00i)T^{2} \)
11 \( 1 + (0.888 + 0.986i)T + (-1.14 + 10.9i)T^{2} \)
13 \( 1 + (-1.79 - 0.912i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (0.585 - 0.722i)T + (-3.53 - 16.6i)T^{2} \)
19 \( 1 + (-4.81 - 2.14i)T + (12.7 + 14.1i)T^{2} \)
23 \( 1 + (0.960 + 0.623i)T + (9.35 + 21.0i)T^{2} \)
29 \( 1 + (4.04 - 5.56i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (9.74 - 1.02i)T + (30.3 - 6.44i)T^{2} \)
37 \( 1 + (-0.401 - 7.65i)T + (-36.7 + 3.86i)T^{2} \)
41 \( 1 + (-0.0622 + 0.0202i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (3.34 + 3.34i)T + 43iT^{2} \)
47 \( 1 + (0.145 - 0.117i)T + (9.77 - 45.9i)T^{2} \)
53 \( 1 + (10.4 + 4.00i)T + (39.3 + 35.4i)T^{2} \)
59 \( 1 + (4.71 + 1.00i)T + (53.8 + 23.9i)T^{2} \)
61 \( 1 + (-0.951 - 4.47i)T + (-55.7 + 24.8i)T^{2} \)
67 \( 1 + (-6.60 - 5.34i)T + (13.9 + 65.5i)T^{2} \)
71 \( 1 + (-8.72 - 6.33i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.64 - 0.0861i)T + (72.6 + 7.63i)T^{2} \)
79 \( 1 + (-10.3 - 1.09i)T + (77.2 + 16.4i)T^{2} \)
83 \( 1 + (-0.978 + 6.17i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + (-15.3 + 3.26i)T + (81.3 - 36.1i)T^{2} \)
97 \( 1 + (-2.72 - 17.1i)T + (-92.2 + 29.9i)T^{2} \)
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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

−12.89590383793083870995095371167, −11.88870523275337039388612956246, −10.76301099840245335840155887502, −9.389998247981883586403015269253, −8.284126993511750145193647263077, −7.69921064273530942891769106816, −6.72142635009885580331616616423, −5.21161586589850371946068123844, −3.69774702702757647685315101983, −1.60632515910715394738086843577, 2.06983385587951446651387685000, 3.39226014384930405785506177108, 5.24438547934150726866184236817, 6.27318895305764801859243389984, 7.59198689923332142847934799762, 9.281203349324206897637248978578, 9.614669755955948902482235806391, 10.83143348408193384050798222160, 11.32386686793847216508293385055, 12.50661385254470068829606793893

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