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Properties

Label 1-14e2-196.127-r1-0-0
Degree $1$
Conductor $196$
Sign $-0.572 - 0.820i$
Analytic cond. $21.0631$
Root an. cond. $21.0631$
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  + (0.900 − 0.433i)3-s + (−0.900 + 0.433i)5-s + (0.623 − 0.781i)9-s + (−0.623 − 0.781i)11-s + (0.623 + 0.781i)13-s + (−0.623 + 0.781i)15-s + (−0.222 − 0.974i)17-s − 19-s + (0.222 − 0.974i)23-s + (0.623 − 0.781i)25-s + (0.222 − 0.974i)27-s + (−0.222 − 0.974i)29-s − 31-s + (−0.900 − 0.433i)33-s + (−0.222 − 0.974i)37-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)3-s + (−0.900 + 0.433i)5-s + (0.623 − 0.781i)9-s + (−0.623 − 0.781i)11-s + (0.623 + 0.781i)13-s + (−0.623 + 0.781i)15-s + (−0.222 − 0.974i)17-s − 19-s + (0.222 − 0.974i)23-s + (0.623 − 0.781i)25-s + (0.222 − 0.974i)27-s + (−0.222 − 0.974i)29-s − 31-s + (−0.900 − 0.433i)33-s + (−0.222 − 0.974i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.572 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.572 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.572 - 0.820i$
Analytic conductor: \(21.0631\)
Root analytic conductor: \(21.0631\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 196,\ (1:\ ),\ -0.572 - 0.820i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6144154729 - 1.177719431i\)
\(L(\frac12)\) \(\approx\) \(0.6144154729 - 1.177719431i\)
\(L(1)\) \(\approx\) \(1.031322483 - 0.3240418538i\)
\(L(1)\) \(\approx\) \(1.031322483 - 0.3240418538i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.900 - 0.433i)T \)
5 \( 1 + (-0.900 + 0.433i)T \)
11 \( 1 + (-0.623 - 0.781i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 + (-0.222 - 0.974i)T \)
19 \( 1 - T \)
23 \( 1 + (0.222 - 0.974i)T \)
29 \( 1 + (-0.222 - 0.974i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.222 - 0.974i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
43 \( 1 + (0.900 + 0.433i)T \)
47 \( 1 + (-0.623 - 0.781i)T \)
53 \( 1 + (-0.222 + 0.974i)T \)
59 \( 1 + (0.900 + 0.433i)T \)
61 \( 1 + (-0.222 - 0.974i)T \)
67 \( 1 - T \)
71 \( 1 + (0.222 - 0.974i)T \)
73 \( 1 + (0.623 - 0.781i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.623 + 0.781i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
97 \( 1 + T \)
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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

−27.36547192659872327019559801739, −25.76813050693921186067460813862, −25.6559024034376969467502096995, −24.142492524341437498860653559329, −23.45859986110124500330356375752, −22.282912334216685226297427010827, −21.09904349685361160908144366311, −20.36579651244330268514806937540, −19.61847262334780736347817124380, −18.70959681660523407589141274841, −17.38931285460657928389232520194, −16.12118177219029959163828953134, −15.34881705562707543975816431812, −14.751971452538730401483066733550, −13.17641920487562592587270196052, −12.659893376292038703333493766280, −11.06951737042613334679864136384, −10.18343737658502638247949796752, −8.83982686822641588909749159587, −8.14161588579281664157813337510, −7.163168584806467657610258405563, −5.30410284963324420391789511853, −4.16918675575237739661255178735, −3.25280942213881446231873178553, −1.69126300779358280931016876481, 0.39681629444658733127715982693, 2.24491678769859681471313375613, 3.35463700497202313105460548203, 4.40495503606128333141823495519, 6.30498499872327843403069802866, 7.30141761606664784192742054860, 8.28446973441073070888462362690, 9.094848881472993537302429558759, 10.6378085168274211315880149720, 11.58933293695329892895281935887, 12.783568866373865031325088692740, 13.769745049641137869334078096301, 14.66664044095100520176926460372, 15.63424812829769896669680384911, 16.50315070807506250222352434127, 18.25818566030721337444425341910, 18.774980458575514620556759401093, 19.57865750777564188688624393519, 20.62901313381340113505827752145, 21.44461241257768593702651820950, 22.84934181743733125350009962973, 23.70519511601669269771354178044, 24.44449680441881608182933998802, 25.581659596878490743761508763456, 26.50639435683421774376961140

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