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Properties

Label 2-175-175.17-c1-0-1
Degree $2$
Conductor $175$
Sign $-0.998 - 0.0463i$
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  + (0.857 − 0.0449i)2-s + (−1.95 + 2.41i)3-s + (−1.25 + 0.131i)4-s + (−0.521 − 2.17i)5-s + (−1.56 + 2.15i)6-s + (−2.09 + 1.61i)7-s + (−2.76 + 0.438i)8-s + (−1.38 − 6.50i)9-s + (−0.544 − 1.84i)10-s + (2.41 + 0.513i)11-s + (2.13 − 3.29i)12-s + (−3.07 + 1.56i)13-s + (−1.72 + 1.47i)14-s + (6.26 + 2.99i)15-s + (0.117 − 0.0249i)16-s + (−1.30 + 3.40i)17-s + ⋯
L(s)  = 1  + (0.606 − 0.0317i)2-s + (−1.12 + 1.39i)3-s + (−0.627 + 0.0659i)4-s + (−0.233 − 0.972i)5-s + (−0.640 + 0.880i)6-s + (−0.793 + 0.608i)7-s + (−0.978 + 0.154i)8-s + (−0.460 − 2.16i)9-s + (−0.172 − 0.582i)10-s + (0.728 + 0.154i)11-s + (0.616 − 0.949i)12-s + (−0.854 + 0.435i)13-s + (−0.461 + 0.394i)14-s + (1.61 + 0.772i)15-s + (0.0293 − 0.00624i)16-s + (−0.316 + 0.825i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0463i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.998 - 0.0463i$
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: \((2,\ 175,\ (\ :1/2),\ -0.998 - 0.0463i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00835839 + 0.360694i\)
\(L(\frac12)\) \(\approx\) \(0.00835839 + 0.360694i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.521 + 2.17i)T \)
7 \( 1 + (2.09 - 1.61i)T \)
good2 \( 1 + (-0.857 + 0.0449i)T + (1.98 - 0.209i)T^{2} \)
3 \( 1 + (1.95 - 2.41i)T + (-0.623 - 2.93i)T^{2} \)
11 \( 1 + (-2.41 - 0.513i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (3.07 - 1.56i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (1.30 - 3.40i)T + (-12.6 - 11.3i)T^{2} \)
19 \( 1 + (0.742 - 7.06i)T + (-18.5 - 3.95i)T^{2} \)
23 \( 1 + (0.107 + 2.04i)T + (-22.8 + 2.40i)T^{2} \)
29 \( 1 + (-0.594 - 0.817i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.48 + 3.32i)T + (-20.7 + 23.0i)T^{2} \)
37 \( 1 + (-0.757 - 0.492i)T + (15.0 + 33.8i)T^{2} \)
41 \( 1 + (-2.23 - 0.727i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-1.54 + 1.54i)T - 43iT^{2} \)
47 \( 1 + (2.51 - 0.965i)T + (34.9 - 31.4i)T^{2} \)
53 \( 1 + (0.683 + 0.553i)T + (11.0 + 51.8i)T^{2} \)
59 \( 1 + (0.825 + 0.917i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (8.07 + 7.26i)T + (6.37 + 60.6i)T^{2} \)
67 \( 1 + (-4.72 - 1.81i)T + (49.7 + 44.8i)T^{2} \)
71 \( 1 + (11.9 - 8.65i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.49 - 6.92i)T + (-29.6 + 66.6i)T^{2} \)
79 \( 1 + (6.17 - 13.8i)T + (-52.8 - 58.7i)T^{2} \)
83 \( 1 + (-0.481 - 3.03i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (6.29 - 6.98i)T + (-9.30 - 88.5i)T^{2} \)
97 \( 1 + (-2.78 + 17.5i)T + (-92.2 - 29.9i)T^{2} \)
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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

−12.69172073578447049971300574925, −12.34874419398906326272886106015, −11.50793088314850862475931836297, −9.989060284236183874299541821012, −9.453094190283840455062574287507, −8.552885834456207763389615512429, −6.22746944501391492080689792432, −5.48912945944509543858765066423, −4.42364687106305354331073520052, −3.77899683311456966752959653427, 0.31025243709242205541084798253, 2.95102831098113879437635654741, 4.68346987071204700833172560918, 5.99968479950292101772761593265, 6.82330341598654033614592597071, 7.48493923686433028456727083717, 9.277927629555489449585438403458, 10.57048011215093599582952427779, 11.56447813369358873983280054848, 12.30176629823803375612245131208

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