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Properties

Label 1-175-175.69-r1-0-0
Degree $1$
Conductor $175$
Sign $-0.968 + 0.248i$
Analytic cond. $18.8063$
Root an. cond. $18.8063$
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  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (−0.309 − 0.951i)6-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.809 + 0.587i)11-s + (−0.809 − 0.587i)12-s + (−0.809 − 0.587i)13-s + (−0.809 − 0.587i)16-s + (0.309 + 0.951i)17-s − 18-s + (−0.309 − 0.951i)19-s + (−0.309 + 0.951i)22-s + (0.809 − 0.587i)23-s − 24-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (−0.309 − 0.951i)6-s + (−0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.809 + 0.587i)11-s + (−0.809 − 0.587i)12-s + (−0.809 − 0.587i)13-s + (−0.809 − 0.587i)16-s + (0.309 + 0.951i)17-s − 18-s + (−0.309 − 0.951i)19-s + (−0.309 + 0.951i)22-s + (0.809 − 0.587i)23-s − 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.968 + 0.248i$
Analytic conductor: \(18.8063\)
Root analytic conductor: \(18.8063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (1:\ ),\ -0.968 + 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2793412540 - 2.211213713i\)
\(L(\frac12)\) \(\approx\) \(-0.2793412540 - 2.211213713i\)
\(L(1)\) \(\approx\) \(0.9797310774 - 1.184291431i\)
\(L(1)\) \(\approx\) \(0.9797310774 - 1.184291431i\)

Euler product

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

−27.178880484002161592313135594284, −26.82369431905847830009661965886, −25.65722531178532421175145850932, −24.98134007713377842602986682275, −23.74909304126232489438985918611, −22.91258975308695114270794563043, −21.840115877062613263824828801517, −21.228015499002592660715296402, −20.39386651883305794579644620588, −19.11053842061737915648206421987, −17.609204359620716740079663766432, −16.37606657588204909373342537910, −16.04674844370984142101337970138, −14.72333936291250566497065246226, −14.19020390160508630990115853534, −13.01685130284560627466091306172, −11.76004187400855318281693629002, −10.69403548459350476552622673027, −9.35689043936077728055825317516, −8.24248934504926529025808568704, −7.140488866928862262846776680895, −5.57406004879732859418849776918, −4.842915839171042503609628537876, −3.56984499213102431096276512965, −2.56525861823557977164422573696, 0.54673098481789006264336517053, 2.12258607380599793134560323214, 2.95453253974928908785301894646, 4.571365927156316647628739660806, 5.79844913390814275973864202251, 6.94541768043181896168036982636, 8.06194373458178843574788798638, 9.593981542178715711470097849257, 10.73988679596894583560218193734, 11.92814479828302976700318363985, 12.87695389072036818787122667980, 13.34558071756135447553992909180, 14.77824234746559799862144342242, 15.22070910675863220819298481004, 17.04533733427600583854743189751, 18.16813961567690789198033977311, 19.128111941880151330975328645214, 19.92526419256610792074658737917, 20.76472377176768139393454427068, 21.87761530810025066295454637326, 23.00518365302669948787128690001, 23.676121092385007665557252954075, 24.571822250531429507052376761243, 25.42944706403535540015218811621, 26.578151001731049295138088628

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