Properties

Label 1-175-175.13-r0-0-0
Degree $1$
Conductor $175$
Sign $0.982 + 0.187i$
Analytic cond. $0.812696$
Root an. cond. $0.812696$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.951 − 0.309i)2-s + (0.587 + 0.809i)3-s + (0.809 − 0.587i)4-s + (0.809 + 0.587i)6-s + (0.587 − 0.809i)8-s + (−0.309 + 0.951i)9-s + (0.309 + 0.951i)11-s + (0.951 + 0.309i)12-s + (−0.951 − 0.309i)13-s + (0.309 − 0.951i)16-s + (0.587 − 0.809i)17-s + i·18-s + (−0.809 − 0.587i)19-s + (0.587 + 0.809i)22-s + (−0.951 + 0.309i)23-s + 24-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (0.587 + 0.809i)3-s + (0.809 − 0.587i)4-s + (0.809 + 0.587i)6-s + (0.587 − 0.809i)8-s + (−0.309 + 0.951i)9-s + (0.309 + 0.951i)11-s + (0.951 + 0.309i)12-s + (−0.951 − 0.309i)13-s + (0.309 − 0.951i)16-s + (0.587 − 0.809i)17-s + i·18-s + (−0.809 − 0.587i)19-s + (0.587 + 0.809i)22-s + (−0.951 + 0.309i)23-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.982 + 0.187i$
Analytic conductor: \(0.812696\)
Root analytic conductor: \(0.812696\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (0:\ ),\ 0.982 + 0.187i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.272815633 + 0.2148443325i\)
\(L(\frac12)\) \(\approx\) \(2.272815633 + 0.2148443325i\)
\(L(1)\) \(\approx\) \(1.988713593 + 0.09698595871i\)
\(L(1)\) \(\approx\) \(1.988713593 + 0.09698595871i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.951 - 0.309i)T \)
3 \( 1 + (0.587 + 0.809i)T \)
11 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (-0.951 - 0.309i)T \)
17 \( 1 + (0.587 - 0.809i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.951 + 0.309i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.587 - 0.809i)T \)
53 \( 1 + (-0.587 - 0.809i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (-0.587 + 0.809i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (0.951 - 0.309i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (-0.587 + 0.809i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.14924462018213692913427557764, −26.057772283114619794268302920596, −25.35147997867857280805942066917, −24.20499320312529385034668142921, −23.97413948359027790024653959916, −22.7072436327604306933419334752, −21.65251322438097432495238081431, −20.79159497079494242452689623422, −19.616961408814601021745630625378, −18.96450606581316805209329608537, −17.427049271536460712345948766336, −16.58271585050957449799681897075, −15.23016429073546519970986396351, −14.31096041510086897879668353233, −13.7200128333520777707172829943, −12.45094364052492590136641773235, −11.95413596942740743622890091109, −10.418177198638513088020260167550, −8.66341559813823002253200037470, −7.82436082531592242954669660570, −6.64485792091609597125126304833, −5.80122804329082864911015137405, −4.161111751455709667382412456697, −3.03497780846917617500535223117, −1.79653092832596328955307676052, 2.08623321940557213981087279511, 3.13551980171833803661258978433, 4.43476840186461125536923165558, 5.11856375570825935010342380983, 6.7186398726964573921882585160, 7.97014058918377996601633171495, 9.64666708050747968830468331530, 10.19415274465554174761911596716, 11.54859903052508344586366031859, 12.52666772658282097251445367294, 13.76188287917822886485942090532, 14.59857619480464016712800366095, 15.353890403892311257588232606140, 16.27879637862939865556818482015, 17.55580276372943876167354530415, 19.31265528674426796671207046570, 19.901469072490898084612948215573, 20.83142817608857146750227912644, 21.659383606762648676383322979317, 22.512113904771316555894184243823, 23.34759535781923596561649595052, 24.75496537973176608986600907137, 25.3167295432202361039037360329, 26.44922138413990911240733037838, 27.68544386706406230307250885771

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