Properties

Label 2-175-5.4-c1-0-9
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  − 2.56i·2-s − 1.56i·3-s − 4.56·4-s − 4·6-s i·7-s + 6.56i·8-s + 0.561·9-s − 1.56·11-s + 7.12i·12-s − 0.438i·13-s − 2.56·14-s + 7.68·16-s − 0.438i·17-s − 1.43i·18-s + 7.12·19-s + ⋯
L(s)  = 1  − 1.81i·2-s − 0.901i·3-s − 2.28·4-s − 1.63·6-s − 0.377i·7-s + 2.31i·8-s + 0.187·9-s − 0.470·11-s + 2.05i·12-s − 0.121i·13-s − 0.684·14-s + 1.92·16-s − 0.106i·17-s − 0.339i·18-s + 1.63·19-s + ⋯

Functional equation

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

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

\(L(1)\) \(\approx\) \(0.227177 + 0.962340i\)
\(L(\frac12)\) \(\approx\) \(0.227177 + 0.962340i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.56iT - 2T^{2} \)
3 \( 1 + 1.56iT - 3T^{2} \)
11 \( 1 + 1.56T + 11T^{2} \)
13 \( 1 + 0.438iT - 13T^{2} \)
17 \( 1 + 0.438iT - 17T^{2} \)
19 \( 1 - 7.12T + 19T^{2} \)
23 \( 1 + 3.12iT - 23T^{2} \)
29 \( 1 + 6.68T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 5.12T + 41T^{2} \)
43 \( 1 + 0.876iT - 43T^{2} \)
47 \( 1 + 8.68iT - 47T^{2} \)
53 \( 1 - 5.12iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 15.3T + 61T^{2} \)
67 \( 1 - 10.2iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 12.2iT - 73T^{2} \)
79 \( 1 - 2.43T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 - 1.12T + 89T^{2} \)
97 \( 1 - 5.80iT - 97T^{2} \)
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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

−12.08763740568024894640142432218, −11.36020954057242596448877493838, −10.29160474035523459066367202414, −9.544419693925570063553528881529, −8.219272527875463299226035180402, −7.11259926744100988530058152192, −5.27504434173653871080047728415, −3.85566179114664969708336788933, −2.47410597584842334148276805115, −1.03512613646503030578168443128, 3.74986861588644646392379912603, 5.02314218023534529060404694834, 5.71978643248983676128468301785, 7.13642732478025094772087550949, 7.944618598253357275994836845905, 9.261338228670521472303535516259, 9.684308185155923627845325571145, 11.13263982060705219395093774094, 12.68691417409336477942535408235, 13.65561250647546665660607995928

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