Properties

Label 2-175-5.4-c1-0-4
Degree $2$
Conductor $175$
Sign $0.894 - 0.447i$
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  + i·3-s + 2·4-s i·7-s + 2·9-s − 3·11-s + 2i·12-s + 5i·13-s + 4·16-s − 3i·17-s − 2·19-s + 21-s − 6i·23-s + 5i·27-s − 2i·28-s − 3·29-s + ⋯
L(s)  = 1  + 0.577i·3-s + 4-s − 0.377i·7-s + 0.666·9-s − 0.904·11-s + 0.577i·12-s + 1.38i·13-s + 16-s − 0.727i·17-s − 0.458·19-s + 0.218·21-s − 1.25i·23-s + 0.962i·27-s − 0.377i·28-s − 0.557·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35728 + 0.320411i\)
\(L(\frac12)\) \(\approx\) \(1.35728 + 0.320411i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good2 \( 1 - 2T^{2} \)
3 \( 1 - iT - 3T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 - 5iT - 13T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 12T + 41T^{2} \)
43 \( 1 + 10iT - 43T^{2} \)
47 \( 1 + 9iT - 47T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 - T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 - iT - 97T^{2} \)
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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

−12.67384449893208941937989550106, −11.67418408201091377603933954843, −10.69041874857904440775009233123, −10.08317058973632407671778685513, −8.815854534325350027084581673107, −7.37276997896353354604911032312, −6.69110116830137008686653654391, −5.12657998933874325446277157686, −3.84327066302090326621443783466, −2.15216858732523752878913131852, 1.81420037246371677867399420653, 3.26198202634456802309882085776, 5.32968870040299501719963145264, 6.36652082627462060432782413242, 7.54287668023440162427597917123, 8.147871851626506410832709692863, 9.898878392443386271295467110125, 10.69582023742138564708576503683, 11.71056418237357816472976395050, 12.86318801592704762278195545106

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