Properties

Label 1-175-175.136-r1-0-0
Degree $1$
Conductor $175$
Sign $-0.895 - 0.445i$
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  + (0.669 + 0.743i)2-s + (0.104 − 0.994i)3-s + (−0.104 + 0.994i)4-s + (0.809 − 0.587i)6-s + (−0.809 + 0.587i)8-s + (−0.978 − 0.207i)9-s + (−0.978 + 0.207i)11-s + (0.978 + 0.207i)12-s + (−0.309 − 0.951i)13-s + (−0.978 − 0.207i)16-s + (−0.913 − 0.406i)17-s + (−0.5 − 0.866i)18-s + (0.104 + 0.994i)19-s + (−0.809 − 0.587i)22-s + (0.669 + 0.743i)23-s + (0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)2-s + (0.104 − 0.994i)3-s + (−0.104 + 0.994i)4-s + (0.809 − 0.587i)6-s + (−0.809 + 0.587i)8-s + (−0.978 − 0.207i)9-s + (−0.978 + 0.207i)11-s + (0.978 + 0.207i)12-s + (−0.309 − 0.951i)13-s + (−0.978 − 0.207i)16-s + (−0.913 − 0.406i)17-s + (−0.5 − 0.866i)18-s + (0.104 + 0.994i)19-s + (−0.809 − 0.587i)22-s + (0.669 + 0.743i)23-s + (0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.895 - 0.445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $-0.895 - 0.445i$
Analytic conductor: \(18.8063\)
Root analytic conductor: \(18.8063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 175,\ (1:\ ),\ -0.895 - 0.445i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02245385048 - 0.09549904896i\)
\(L(\frac12)\) \(\approx\) \(0.02245385048 - 0.09549904896i\)
\(L(1)\) \(\approx\) \(0.9957500551 + 0.1416601807i\)
\(L(1)\) \(\approx\) \(0.9957500551 + 0.1416601807i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.669 - 0.743i)T \)
3 \( 1 + (-0.104 + 0.994i)T \)
11 \( 1 + (0.978 - 0.207i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (0.913 + 0.406i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (-0.669 - 0.743i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (0.978 + 0.207i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 - T \)
47 \( 1 + (0.913 - 0.406i)T \)
53 \( 1 + (0.104 - 0.994i)T \)
59 \( 1 + (0.669 - 0.743i)T \)
61 \( 1 + (0.669 + 0.743i)T \)
67 \( 1 + (-0.913 - 0.406i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (-0.978 + 0.207i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (0.669 + 0.743i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−27.81020216457614549170730189093, −26.70187221737726352878040634349, −26.00806619903076883650282388367, −24.41494917765541460730869349537, −23.61204113532055945874125320805, −22.4794152094672626382290734165, −21.730623669726633253661078338492, −21.01685665193904481983040966130, −20.10531076455133278353012358624, −19.21013343193051351013110677837, −18.00125298890119825150948885624, −16.57072905891166763415573683876, −15.54958364591423178755101643614, −14.7425205186308398791657720291, −13.708320178078135420180693863353, −12.69933103114991553833785869096, −11.2625342023004436850240748460, −10.766013979742315641287981856035, −9.559590776727817039478314136460, −8.68528540268237905575659388449, −6.73028830573017685108418451155, −5.276141420602944560421443181546, −4.54267715974020927544673947725, −3.30365790989987669473854298800, −2.17654467911369318258747648031, 0.02491818156008686788167878168, 2.218709482895168813476095046616, 3.41025456135578476176775371298, 5.12924213504328728334302829597, 5.98418397982518445804330772683, 7.3373092611785756618308964215, 7.84095046069108352095060443946, 9.11652708454612101976524354325, 10.93441536022005095067594432999, 12.211689333583327326277588426607, 12.99010480360816138463251800801, 13.725922475313542579606126896212, 14.88202708635595144805653467818, 15.743409479100942694675695965899, 17.080384483872043512477066244003, 17.84977305421442383123282143394, 18.75228089074859254055944413970, 20.17008638988022404444593926671, 20.97083020363735049575562801605, 22.4488537923333702773999902788, 23.01914394288078648412016119324, 24.03748061534014411101628374700, 24.7378585499409307416782514555, 25.57256330450464056061945388973, 26.40776580850533784135679694961

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