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Properties

Label 2-175-175.47-c1-0-5
Degree $2$
Conductor $175$
Sign $-0.661 - 0.749i$
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  + (−0.0825 + 1.57i)2-s + (−0.0997 + 0.0808i)3-s + (−0.483 − 0.0508i)4-s + (−1.93 + 1.11i)5-s + (−0.118 − 0.163i)6-s + (2.63 − 0.202i)7-s + (−0.373 + 2.35i)8-s + (−0.620 + 2.91i)9-s + (−1.59 − 3.14i)10-s + (−3.49 + 0.743i)11-s + (0.0523 − 0.0339i)12-s + (0.403 − 0.791i)13-s + (0.101 + 4.17i)14-s + (0.103 − 0.267i)15-s + (−4.63 − 0.984i)16-s + (6.15 − 2.36i)17-s + ⋯
L(s)  = 1  + (−0.0583 + 1.11i)2-s + (−0.0576 + 0.0466i)3-s + (−0.241 − 0.0254i)4-s + (−0.866 + 0.499i)5-s + (−0.0485 − 0.0668i)6-s + (0.997 − 0.0765i)7-s + (−0.132 + 0.833i)8-s + (−0.206 + 0.972i)9-s + (−0.505 − 0.993i)10-s + (−1.05 + 0.224i)11-s + (0.0151 − 0.00981i)12-s + (0.111 − 0.219i)13-s + (0.0270 + 1.11i)14-s + (0.0266 − 0.0691i)15-s + (−1.15 − 0.246i)16-s + (1.49 − 0.572i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.661 - 0.749i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ -0.661 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.438507 + 0.971756i\)
\(L(\frac12)\) \(\approx\) \(0.438507 + 0.971756i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.93 - 1.11i)T \)
7 \( 1 + (-2.63 + 0.202i)T \)
good2 \( 1 + (0.0825 - 1.57i)T + (-1.98 - 0.209i)T^{2} \)
3 \( 1 + (0.0997 - 0.0808i)T + (0.623 - 2.93i)T^{2} \)
11 \( 1 + (3.49 - 0.743i)T + (10.0 - 4.47i)T^{2} \)
13 \( 1 + (-0.403 + 0.791i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (-6.15 + 2.36i)T + (12.6 - 11.3i)T^{2} \)
19 \( 1 + (0.107 + 1.02i)T + (-18.5 + 3.95i)T^{2} \)
23 \( 1 + (-2.68 - 0.140i)T + (22.8 + 2.40i)T^{2} \)
29 \( 1 + (0.242 - 0.334i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.79 + 4.02i)T + (-20.7 - 23.0i)T^{2} \)
37 \( 1 + (-5.67 - 8.74i)T + (-15.0 + 33.8i)T^{2} \)
41 \( 1 + (0.884 - 0.287i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (-3.39 + 3.39i)T - 43iT^{2} \)
47 \( 1 + (-2.43 + 6.33i)T + (-34.9 - 31.4i)T^{2} \)
53 \( 1 + (5.17 + 6.38i)T + (-11.0 + 51.8i)T^{2} \)
59 \( 1 + (-3.83 + 4.25i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (3.99 - 3.59i)T + (6.37 - 60.6i)T^{2} \)
67 \( 1 + (-3.90 - 10.1i)T + (-49.7 + 44.8i)T^{2} \)
71 \( 1 + (-5.23 - 3.80i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (9.62 + 6.25i)T + (29.6 + 66.6i)T^{2} \)
79 \( 1 + (5.43 + 12.2i)T + (-52.8 + 58.7i)T^{2} \)
83 \( 1 + (10.5 + 1.67i)T + (78.9 + 25.6i)T^{2} \)
89 \( 1 + (-3.49 - 3.88i)T + (-9.30 + 88.5i)T^{2} \)
97 \( 1 + (14.9 - 2.36i)T + (92.2 - 29.9i)T^{2} \)
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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

−13.37279906005886106468150311906, −11.83707346695819399686556554786, −11.17102621183245438560379269150, −10.24147895161352968322093684481, −8.338532942096355084417170902184, −7.83477511630986241166786312720, −7.17128778178475511638814986527, −5.56501810626016505867541505376, −4.75826800579302726028119731342, −2.70784218509486283743422208273, 1.13434108344645080041028676974, 3.05991778022188021729141635215, 4.25080564090156924566601077570, 5.70782975799851445230582793175, 7.40901232207654782296715275805, 8.362224804597699335190724055186, 9.492482754637843378385169209461, 10.74362333400670285490031529673, 11.34942630966504280122967945212, 12.34224031183747493377354737373

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