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Properties

Label 1-14e2-196.143-r0-0-0
Degree $1$
Conductor $196$
Sign $-0.905 + 0.424i$
Analytic cond. $0.910220$
Root an. cond. $0.910220$
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  + (0.0747 + 0.997i)3-s + (−0.826 + 0.563i)5-s + (−0.988 + 0.149i)9-s + (0.988 + 0.149i)11-s + (−0.623 + 0.781i)13-s + (−0.623 − 0.781i)15-s + (−0.955 + 0.294i)17-s + (−0.5 + 0.866i)19-s + (−0.955 − 0.294i)23-s + (0.365 − 0.930i)25-s + (−0.222 − 0.974i)27-s + (−0.222 + 0.974i)29-s + (−0.5 − 0.866i)31-s + (−0.0747 + 0.997i)33-s + (−0.733 − 0.680i)37-s + ⋯
L(s)  = 1  + (0.0747 + 0.997i)3-s + (−0.826 + 0.563i)5-s + (−0.988 + 0.149i)9-s + (0.988 + 0.149i)11-s + (−0.623 + 0.781i)13-s + (−0.623 − 0.781i)15-s + (−0.955 + 0.294i)17-s + (−0.5 + 0.866i)19-s + (−0.955 − 0.294i)23-s + (0.365 − 0.930i)25-s + (−0.222 − 0.974i)27-s + (−0.222 + 0.974i)29-s + (−0.5 − 0.866i)31-s + (−0.0747 + 0.997i)33-s + (−0.733 − 0.680i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.905 + 0.424i$
Analytic conductor: \(0.910220\)
Root analytic conductor: \(0.910220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 196,\ (0:\ ),\ -0.905 + 0.424i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1557492863 + 0.6995909135i\)
\(L(\frac12)\) \(\approx\) \(0.1557492863 + 0.6995909135i\)
\(L(1)\) \(\approx\) \(0.6666228867 + 0.4573820038i\)
\(L(1)\) \(\approx\) \(0.6666228867 + 0.4573820038i\)

Euler product

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

−26.60732357080789878994531374370, −25.38391897198453731183254013043, −24.48038532117976383604528187488, −24.01646843078891852564396437978, −22.88514637003283912878551701908, −22.08767399194110169651279117037, −20.46721578488211478438621690798, −19.70241360574104989639267760302, −19.21872473243222115844037294013, −17.787772986572149278279744470339, −17.17361942364458176598315493788, −15.891731714761234483637992624867, −14.862202780302886845779900861510, −13.7135009127549817139533879610, −12.7137998186833636265809943213, −11.93322540553644138791993090868, −11.058537128780478152278641762527, −9.27940609916395047728561577448, −8.36409668985034972694905672993, −7.39906130462142198666474538240, −6.39268023189699691695416241741, −4.9760243673152301870674810591, −3.62402989384936281835226594715, −2.127913554816526014007579430287, −0.53384137687485752868613810067, 2.31156045216542106386227472143, 3.87252345229320936244348421945, 4.31783342423934434239858196866, 6.00284041493556185916029054470, 7.18815607473843563685563644822, 8.50726539687248855658308209952, 9.464978945563252209617269803292, 10.62086522912785033981931180327, 11.45097963336826931962947292274, 12.40620093272988763419633483532, 14.2977687977152328432830160728, 14.63873504717817629777284340961, 15.783785095839797071769929157436, 16.58058669801024051263711149012, 17.6192342258134249626582340688, 19.04885147317704410261752354361, 19.76088378243081499536357379087, 20.67116183920535533014975797272, 22.0515985312629190445374189357, 22.291901112479469137673421426422, 23.480199816440109493995471183876, 24.516723417083019697537253686785, 25.84477525159976040219187403048, 26.51793726027772003056016520122, 27.385672109561240950309752775399

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