Properties

Label 2-175-175.103-c1-0-4
Degree $2$
Conductor $175$
Sign $0.321 - 0.946i$
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  + (0.000942 + 4.94e−5i)2-s + (1.67 + 2.07i)3-s + (−1.98 − 0.209i)4-s + (2.16 + 0.564i)5-s + (0.00147 + 0.00203i)6-s + (−2.21 + 1.45i)7-s + (−0.00372 − 0.000590i)8-s + (−0.853 + 4.01i)9-s + (0.00201 + 0.000639i)10-s + (4.83 − 1.02i)11-s + (−2.90 − 4.47i)12-s + (−4.82 − 2.45i)13-s + (−0.00215 + 0.00126i)14-s + (2.45 + 5.43i)15-s + (3.91 + 0.831i)16-s + (0.276 + 0.720i)17-s + ⋯
L(s)  = 1  + (0.000666 + 3.49e−5i)2-s + (0.968 + 1.19i)3-s + (−0.994 − 0.104i)4-s + (0.967 + 0.252i)5-s + (0.000603 + 0.000831i)6-s + (−0.835 + 0.549i)7-s + (−0.00131 − 0.000208i)8-s + (−0.284 + 1.33i)9-s + (0.000636 + 0.000202i)10-s + (1.45 − 0.309i)11-s + (−0.838 − 1.29i)12-s + (−1.33 − 0.681i)13-s + (−0.000576 + 0.000337i)14-s + (0.635 + 1.40i)15-s + (0.978 + 0.207i)16-s + (0.0671 + 0.174i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.321 - 0.946i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ 0.321 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11072 + 0.795517i\)
\(L(\frac12)\) \(\approx\) \(1.11072 + 0.795517i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.16 - 0.564i)T \)
7 \( 1 + (2.21 - 1.45i)T \)
good2 \( 1 + (-0.000942 - 4.94e-5i)T + (1.98 + 0.209i)T^{2} \)
3 \( 1 + (-1.67 - 2.07i)T + (-0.623 + 2.93i)T^{2} \)
11 \( 1 + (-4.83 + 1.02i)T + (10.0 - 4.47i)T^{2} \)
13 \( 1 + (4.82 + 2.45i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (-0.276 - 0.720i)T + (-12.6 + 11.3i)T^{2} \)
19 \( 1 + (-0.101 - 0.962i)T + (-18.5 + 3.95i)T^{2} \)
23 \( 1 + (-0.143 + 2.73i)T + (-22.8 - 2.40i)T^{2} \)
29 \( 1 + (-2.32 + 3.19i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (-3.67 + 8.24i)T + (-20.7 - 23.0i)T^{2} \)
37 \( 1 + (1.77 - 1.15i)T + (15.0 - 33.8i)T^{2} \)
41 \( 1 + (6.23 - 2.02i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (2.74 + 2.74i)T + 43iT^{2} \)
47 \( 1 + (4.18 + 1.60i)T + (34.9 + 31.4i)T^{2} \)
53 \( 1 + (0.179 - 0.145i)T + (11.0 - 51.8i)T^{2} \)
59 \( 1 + (-2.95 + 3.28i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (9.04 - 8.14i)T + (6.37 - 60.6i)T^{2} \)
67 \( 1 + (-0.672 + 0.258i)T + (49.7 - 44.8i)T^{2} \)
71 \( 1 + (-3.49 - 2.53i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.442 + 0.681i)T + (-29.6 - 66.6i)T^{2} \)
79 \( 1 + (-2.01 - 4.51i)T + (-52.8 + 58.7i)T^{2} \)
83 \( 1 + (2.09 - 13.2i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + (7.76 + 8.62i)T + (-9.30 + 88.5i)T^{2} \)
97 \( 1 + (-2.13 - 13.4i)T + (-92.2 + 29.9i)T^{2} \)
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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

−13.21235431491799340277910220522, −12.07854621530704978606393606269, −10.23335210619195075195767113533, −9.768125700263175605949999152946, −9.204002453298126265220466463200, −8.310885016782267738125743811074, −6.38305369686292704477704362347, −5.15946914647570019248208228475, −3.92241820065715560134713296203, −2.75663798928704502256733736974, 1.48641406678540100449583055191, 3.19766433205701832340081255744, 4.76278991691560520784956229798, 6.53766898517573956048095796399, 7.20961152024508473874024893795, 8.651829798671872737714679875864, 9.347857519982908611151133730648, 9.992740885701981033355439686451, 12.17609185284501537289434622621, 12.65032967101002199228570278501

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