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Properties

Label 2-175-35.9-c1-0-4
Degree $2$
Conductor $175$
Sign $0.374 - 0.927i$
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  + (2.09 + 1.20i)2-s + (−0.358 + 0.207i)3-s + (1.91 + 3.31i)4-s − 6-s + (−0.358 − 2.62i)7-s + 4.41i·8-s + (−1.41 + 2.44i)9-s + (0.414 + 0.717i)11-s + (−1.37 − 0.792i)12-s − 4.82i·13-s + (2.41 − 5.91i)14-s + (−1.49 + 2.59i)16-s + (−4.18 + 2.41i)17-s + (−5.91 + 3.41i)18-s + (1.41 − 2.44i)19-s + ⋯
L(s)  = 1  + (1.47 + 0.853i)2-s + (−0.207 + 0.119i)3-s + (0.957 + 1.65i)4-s − 0.408·6-s + (−0.135 − 0.990i)7-s + 1.56i·8-s + (−0.471 + 0.816i)9-s + (0.124 + 0.216i)11-s + (−0.396 − 0.228i)12-s − 1.33i·13-s + (0.645 − 1.58i)14-s + (−0.374 + 0.649i)16-s + (−1.01 + 0.585i)17-s + (−1.39 + 0.804i)18-s + (0.324 − 0.561i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.374 - 0.927i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ 0.374 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79901 + 1.21319i\)
\(L(\frac12)\) \(\approx\) \(1.79901 + 1.21319i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.358 + 2.62i)T \)
good2 \( 1 + (-2.09 - 1.20i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.358 - 0.207i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-0.414 - 0.717i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.82iT - 13T^{2} \)
17 \( 1 + (4.18 - 2.41i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.41 + 2.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.358 + 0.207i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.82T + 41T^{2} \)
43 \( 1 - 3.58iT - 43T^{2} \)
47 \( 1 + (-1.73 - i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.01 - 0.585i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.24 - 3.88i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.74 - 4.75i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.30 - 4.79i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.48T + 71T^{2} \)
73 \( 1 + (0.717 - 0.414i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.41 + 12.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.7iT - 83T^{2} \)
89 \( 1 + (4.32 - 7.49i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.6iT - 97T^{2} \)
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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

−13.32423287510966803561406591107, −12.29366194952513095263657396146, −11.09108149590536382747889694081, −10.23621517230577354850351454351, −8.354329359010799927633842627509, −7.38971195077287413445759617840, −6.40441048761140737916275555730, −5.25666793200693757427126410340, −4.37273869551984358665684694296, −3.02878436875264574413521983813, 2.13208403683115792993625752563, 3.45056157374777453049629116384, 4.72332895102875896868947592541, 5.90498489555092274008799387903, 6.65422505722871314480614206509, 8.725786093022766177875049841664, 9.701192776501904289175719571678, 11.20896857936598448924597509421, 11.77645059425982187009166164087, 12.31417631974415999563332560831

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