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Properties

Label 2-175-5.4-c3-0-15
Degree $2$
Conductor $175$
Sign $0.894 + 0.447i$
Analytic cond. $10.3253$
Root an. cond. $3.21330$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.58i·2-s − 6.65i·3-s + 1.31·4-s + 17.2·6-s − 7i·7-s + 24.0i·8-s − 17.3·9-s + 38.2·11-s − 8.74i·12-s − 19.3i·13-s + 18.1·14-s − 51.7·16-s − 87.2i·17-s − 44.7i·18-s + 44.2·19-s + ⋯
L(s)  = 1  + 0.914i·2-s − 1.28i·3-s + 0.164·4-s + 1.17·6-s − 0.377i·7-s + 1.06i·8-s − 0.641·9-s + 1.04·11-s − 0.210i·12-s − 0.412i·13-s + 0.345·14-s − 0.808·16-s − 1.24i·17-s − 0.586i·18-s + 0.534·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(10.3253\)
Root analytic conductor: \(3.21330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :3/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.96631 - 0.464183i\)
\(L(\frac12)\) \(\approx\) \(1.96631 - 0.464183i\)
\(L(\frac{5}{2})\) 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 \( 1 + 7iT \)
good2 \( 1 - 2.58iT - 8T^{2} \)
3 \( 1 + 6.65iT - 27T^{2} \)
11 \( 1 - 38.2T + 1.33e3T^{2} \)
13 \( 1 + 19.3iT - 2.19e3T^{2} \)
17 \( 1 + 87.2iT - 4.91e3T^{2} \)
19 \( 1 - 44.2T + 6.85e3T^{2} \)
23 \( 1 + 218. iT - 1.21e4T^{2} \)
29 \( 1 - 46.9T + 2.43e4T^{2} \)
31 \( 1 - 194.T + 2.97e4T^{2} \)
37 \( 1 - 366. iT - 5.06e4T^{2} \)
41 \( 1 + 339.T + 6.89e4T^{2} \)
43 \( 1 - 226. iT - 7.95e4T^{2} \)
47 \( 1 - 11.6iT - 1.03e5T^{2} \)
53 \( 1 - 209. iT - 1.48e5T^{2} \)
59 \( 1 - 616T + 2.05e5T^{2} \)
61 \( 1 - 320.T + 2.26e5T^{2} \)
67 \( 1 - 14.5iT - 3.00e5T^{2} \)
71 \( 1 + 952T + 3.57e5T^{2} \)
73 \( 1 + 824. iT - 3.89e5T^{2} \)
79 \( 1 + 156.T + 4.93e5T^{2} \)
83 \( 1 - 1.03e3iT - 5.71e5T^{2} \)
89 \( 1 - 170.T + 7.04e5T^{2} \)
97 \( 1 - 1.05e3iT - 9.12e5T^{2} \)
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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

−12.10399213872292715096570306755, −11.59700987177039495360289214855, −10.15404941878339276654717952647, −8.624255205988623097960181033632, −7.77298554987503961263365679156, −6.77644020452773905440791942022, −6.40188160329373191011090938787, −4.82352917790929012541746536107, −2.67352045577803876477612793986, −1.03237242944832544739333277365, 1.64671238278383611027339063637, 3.40688579798447119643912695672, 4.13944779435838243460918755144, 5.68141460145288017547113342330, 7.02987950355812412862414971156, 8.758874945010871124384702912915, 9.664806314256019606592320395801, 10.28463169139887517179348182900, 11.38732023366106231059617142126, 11.88224123264825821749954534786

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