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Properties

Label 2-175-35.17-c1-0-3
Degree $2$
Conductor $175$
Sign $0.0932 - 0.995i$
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  + (−0.5 + 1.86i)2-s + (1.86 − 0.5i)3-s + (−1.5 − 0.866i)4-s + 3.73i·6-s + (2.5 − 0.866i)7-s + (−0.366 + 0.366i)8-s + (0.633 − 0.366i)9-s + (−0.366 + 0.633i)11-s + (−3.23 − 0.866i)12-s + (2 + 2i)13-s + (0.366 + 5.09i)14-s + (−2.23 − 3.86i)16-s + (−0.267 − i)17-s + (0.366 + 1.36i)18-s + (−1.36 − 2.36i)19-s + ⋯
L(s)  = 1  + (−0.353 + 1.31i)2-s + (1.07 − 0.288i)3-s + (−0.750 − 0.433i)4-s + 1.52i·6-s + (0.944 − 0.327i)7-s + (−0.129 + 0.129i)8-s + (0.211 − 0.122i)9-s + (−0.110 + 0.191i)11-s + (−0.933 − 0.249i)12-s + (0.554 + 0.554i)13-s + (0.0978 + 1.36i)14-s + (−0.558 − 0.966i)16-s + (−0.0649 − 0.242i)17-s + (0.0862 + 0.321i)18-s + (−0.313 − 0.542i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0932 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0932 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.0932 - 0.995i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ 0.0932 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00922 + 0.919148i\)
\(L(\frac12)\) \(\approx\) \(1.00922 + 0.919148i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + (-2.5 + 0.866i)T \)
good2 \( 1 + (0.5 - 1.86i)T + (-1.73 - i)T^{2} \)
3 \( 1 + (-1.86 + 0.5i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (0.366 - 0.633i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2 - 2i)T + 13iT^{2} \)
17 \( 1 + (0.267 + i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.36 + 2.36i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.96 + 1.86i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 + (-0.464 - 0.267i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.26 + 4.73i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 0.464iT - 41T^{2} \)
43 \( 1 + (-5.83 + 5.83i)T - 43iT^{2} \)
47 \( 1 + (0.633 + 0.169i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.83 - 6.83i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.09 + 1.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.33 - 4.23i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.13 - 0.303i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 4.73T + 71T^{2} \)
73 \( 1 + (3.46 - 0.928i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.83 - 3.36i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.09 + 3.09i)T + 83iT^{2} \)
89 \( 1 + (-8.33 - 14.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.92 - 7.92i)T - 97iT^{2} \)
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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

−13.57805011234102473125936657435, −12.01392047167244975354353624836, −10.88644347084466343997966151868, −9.359998870139710593502553519480, −8.514992574883318586313382205631, −7.87213923219163673730412042619, −7.03387974945067847873161206004, −5.74758904399241084903752998965, −4.26756424717461960487827618232, −2.29619298643033443023553836225, 1.79179775446068053194037104892, 3.03844382634343125327799957583, 4.13480834724042785373400456560, 5.94698876046510934285152931536, 7.992302096173380204486950388917, 8.571087086125865018280357885223, 9.577978688692028815821798480493, 10.49485796838929018920752595702, 11.39375081817235665786431085486, 12.26441527082804776198004296263

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