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Properties

Label 2-175-35.3-c1-0-5
Degree $2$
Conductor $175$
Sign $0.528 - 0.848i$
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  + (1.86 + 0.5i)2-s + (0.5 + 1.86i)3-s + (1.5 + 0.866i)4-s + 3.73i·6-s + (−0.866 − 2.5i)7-s + (−0.366 − 0.366i)8-s + (−0.633 + 0.366i)9-s + (−0.366 + 0.633i)11-s + (−0.866 + 3.23i)12-s + (−2 + 2i)13-s + (−0.366 − 5.09i)14-s + (−2.23 − 3.86i)16-s + (−1 + 0.267i)17-s + (−1.36 + 0.366i)18-s + (1.36 + 2.36i)19-s + ⋯
L(s)  = 1  + (1.31 + 0.353i)2-s + (0.288 + 1.07i)3-s + (0.750 + 0.433i)4-s + 1.52i·6-s + (−0.327 − 0.944i)7-s + (−0.129 − 0.129i)8-s + (−0.211 + 0.122i)9-s + (−0.110 + 0.191i)11-s + (−0.249 + 0.933i)12-s + (−0.554 + 0.554i)13-s + (−0.0978 − 1.36i)14-s + (−0.558 − 0.966i)16-s + (−0.242 + 0.0649i)17-s + (−0.321 + 0.0862i)18-s + (0.313 + 0.542i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.528 - 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.528 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.528 - 0.848i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ 0.528 - 0.848i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89477 + 1.05215i\)
\(L(\frac12)\) \(\approx\) \(1.89477 + 1.05215i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 2.5i)T \)
good2 \( 1 + (-1.86 - 0.5i)T + (1.73 + i)T^{2} \)
3 \( 1 + (-0.5 - 1.86i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (0.366 - 0.633i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2 - 2i)T - 13iT^{2} \)
17 \( 1 + (1 - 0.267i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.36 - 2.36i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.86 + 6.96i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 3iT - 29T^{2} \)
31 \( 1 + (-0.464 - 0.267i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.73 + 1.26i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.464iT - 41T^{2} \)
43 \( 1 + (-5.83 - 5.83i)T + 43iT^{2} \)
47 \( 1 + (0.169 - 0.633i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.83 - 1.83i)T + (45.8 - 26.5i)T^{2} \)
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 \( 1 + (-0.303 - 1.13i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 4.73T + 71T^{2} \)
73 \( 1 + (0.928 + 3.46i)T + (-63.2 + 36.5i)T^{2} \)
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

−12.99745631135251374435762322467, −12.25448940806984871885133813109, −10.83549863918711842637830735410, −9.959099192105244060660148450271, −9.046183783987994648384291707190, −7.34447332494334016284520953283, −6.36236822680905203260413741577, −4.84279489707216291796343415807, −4.23881748632353773422303780243, −3.15469266268573948495990219819, 2.19408184833160732483297456449, 3.25084719096874303214038866598, 4.98589100177498931811556459960, 5.94905406398640255453906854376, 7.12724396592915383925060826454, 8.313065062143812665050290758394, 9.500717930306257253046347502308, 11.10391794969813144605160391994, 12.09018640147272733531022263743, 12.62054051266904160859215183742

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