Properties

Label 2-175-25.6-c1-0-10
Degree $2$
Conductor $175$
Sign $0.874 - 0.485i$
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  + (0.511 + 1.57i)2-s + (1.62 − 1.17i)3-s + (−0.595 + 0.432i)4-s + (−0.827 − 2.07i)5-s + (2.68 + 1.94i)6-s − 7-s + (1.69 + 1.22i)8-s + (0.315 − 0.969i)9-s + (2.84 − 2.36i)10-s + (0.651 + 2.00i)11-s + (−0.456 + 1.40i)12-s + (−0.200 + 0.616i)13-s + (−0.511 − 1.57i)14-s + (−3.79 − 2.39i)15-s + (−1.52 + 4.68i)16-s + (−0.514 − 0.373i)17-s + ⋯
L(s)  = 1  + (0.361 + 1.11i)2-s + (0.936 − 0.680i)3-s + (−0.297 + 0.216i)4-s + (−0.370 − 0.928i)5-s + (1.09 + 0.795i)6-s − 0.377·7-s + (0.597 + 0.434i)8-s + (0.105 − 0.323i)9-s + (0.899 − 0.747i)10-s + (0.196 + 0.604i)11-s + (−0.131 + 0.405i)12-s + (−0.0555 + 0.171i)13-s + (−0.136 − 0.420i)14-s + (−0.978 − 0.618i)15-s + (−0.380 + 1.17i)16-s + (−0.124 − 0.0905i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.874 - 0.485i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ 0.874 - 0.485i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67759 + 0.434584i\)
\(L(\frac12)\) \(\approx\) \(1.67759 + 0.434584i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.827 + 2.07i)T \)
7 \( 1 + T \)
good2 \( 1 + (-0.511 - 1.57i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-1.62 + 1.17i)T + (0.927 - 2.85i)T^{2} \)
11 \( 1 + (-0.651 - 2.00i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.200 - 0.616i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.514 + 0.373i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (5.01 + 3.64i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.0164 - 0.0507i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (3.40 - 2.47i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.94 + 1.41i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.55 + 4.77i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.851 - 2.61i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 3.21T + 43T^{2} \)
47 \( 1 + (-10.2 + 7.45i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-9.67 + 7.03i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.22 - 12.9i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.26 + 3.90i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (4.78 + 3.47i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-12.9 + 9.42i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.69 + 5.21i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (12.4 - 9.01i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-13.6 - 9.88i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-2.55 - 7.87i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-6.99 + 5.08i)T + (29.9 - 92.2i)T^{2} \)
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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

−13.16787914754177363682985442917, −12.26629579279165100787278702287, −10.86860264108703425326590538145, −9.220649480174643413285343070065, −8.473125581904216327167114950688, −7.50375321241986301707197615835, −6.77186575780001441975289801389, −5.34644745243340974234829423646, −4.14421633496996967541670557432, −2.07216769025241541113197916122, 2.46396330338223059275581980275, 3.46232628197068558879925494869, 4.13927558339510315240380243500, 6.26586487290768953916946817718, 7.62709561778390252066185607699, 8.819617877340884381476953122352, 10.00286898051046733073130366823, 10.60933690568479314703965575238, 11.55415724090512808315100020179, 12.53197009289487239624347176132

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