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Properties

Label 1-14e2-196.39-r1-0-0
Degree $1$
Conductor $196$
Sign $0.891 + 0.453i$
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  + (−0.365 + 0.930i)3-s + (−0.988 + 0.149i)5-s + (−0.733 − 0.680i)9-s + (0.733 − 0.680i)11-s + (−0.222 − 0.974i)13-s + (0.222 − 0.974i)15-s + (0.0747 + 0.997i)17-s + (0.5 − 0.866i)19-s + (−0.0747 + 0.997i)23-s + (0.955 − 0.294i)25-s + (0.900 − 0.433i)27-s + (−0.900 − 0.433i)29-s + (0.5 + 0.866i)31-s + (0.365 + 0.930i)33-s + (0.826 − 0.563i)37-s + ⋯
L(s)  = 1  + (−0.365 + 0.930i)3-s + (−0.988 + 0.149i)5-s + (−0.733 − 0.680i)9-s + (0.733 − 0.680i)11-s + (−0.222 − 0.974i)13-s + (0.222 − 0.974i)15-s + (0.0747 + 0.997i)17-s + (0.5 − 0.866i)19-s + (−0.0747 + 0.997i)23-s + (0.955 − 0.294i)25-s + (0.900 − 0.433i)27-s + (−0.900 − 0.433i)29-s + (0.5 + 0.866i)31-s + (0.365 + 0.930i)33-s + (0.826 − 0.563i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.891 + 0.453i$
Analytic conductor: \(21.0631\)
Root analytic conductor: \(21.0631\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 196,\ (1:\ ),\ 0.891 + 0.453i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.166110966 + 0.2792996129i\)
\(L(\frac12)\) \(\approx\) \(1.166110966 + 0.2792996129i\)
\(L(1)\) \(\approx\) \(0.8273001503 + 0.1811840721i\)
\(L(1)\) \(\approx\) \(0.8273001503 + 0.1811840721i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.365 + 0.930i)T \)
5 \( 1 + (-0.988 + 0.149i)T \)
11 \( 1 + (0.733 - 0.680i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (0.0747 + 0.997i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.0747 + 0.997i)T \)
29 \( 1 + (-0.900 - 0.433i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.826 - 0.563i)T \)
41 \( 1 + (0.623 + 0.781i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
47 \( 1 + (-0.955 - 0.294i)T \)
53 \( 1 + (0.826 + 0.563i)T \)
59 \( 1 + (0.988 + 0.149i)T \)
61 \( 1 + (0.826 - 0.563i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (0.955 - 0.294i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (-0.733 - 0.680i)T \)
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

−26.787059848812427112985918208956, −25.565877109545908060699282399989, −24.52810913700428918724512040668, −23.979940923833741280576647706191, −22.79371841394973892263046571402, −22.464571381334426101752806660179, −20.68277740332758623334998884846, −19.87560895891210010441722041556, −18.908004527788675804401565731192, −18.26110820349522396899099686949, −16.88872476176403383089037123533, −16.32269994633851720639606160, −14.85900697921954969625071863237, −13.984332232993354483826659726389, −12.6633390835132228555103401599, −11.89192957808369703523861666698, −11.284564307696426683231127796012, −9.63792665956622722497103529104, −8.37067683209800695761469118979, −7.333131600809075745697537244295, −6.61142329061024482658193737169, −5.08684717036891920756716772257, −3.89581752766682015354723422932, −2.22057845428618504780341460071, −0.788050295360059478872826843198, 0.696246239103063253262169218120, 3.14153171876136634477530421573, 3.92514416377923000002622706144, 5.15932969591582036108609231939, 6.32557510733513289891572074432, 7.771235351826238195627474086825, 8.82245581906435456580961076274, 9.98285013149463943373181172958, 11.13102558818373547582542425246, 11.66704261898831851709783459907, 12.96965849875302568722524920057, 14.50911728893069806970291104953, 15.25731686656675998680149435169, 16.069863030313316844059517321577, 17.02668547064937016028588214622, 18.00641523093752053678638974681, 19.543483351803390759925153856857, 19.9208240231943327494822827188, 21.30284726203822986180787469048, 22.09609355923404520719626241778, 22.928127654144071507105680380513, 23.78837070077820784163022561487, 24.88505771390736647047053128482, 26.259459491798742158075870630260, 26.854116734570365902136585790777

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