Properties

Label 1-14e2-196.187-r0-0-0
Degree $1$
Conductor $196$
Sign $0.999 + 0.0320i$
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  + (0.955 + 0.294i)3-s + (0.733 − 0.680i)5-s + (0.826 + 0.563i)9-s + (−0.826 + 0.563i)11-s + (0.900 + 0.433i)13-s + (0.900 − 0.433i)15-s + (−0.365 − 0.930i)17-s + (−0.5 − 0.866i)19-s + (−0.365 + 0.930i)23-s + (0.0747 − 0.997i)25-s + (0.623 + 0.781i)27-s + (0.623 − 0.781i)29-s + (−0.5 + 0.866i)31-s + (−0.955 + 0.294i)33-s + (−0.988 − 0.149i)37-s + ⋯
L(s)  = 1  + (0.955 + 0.294i)3-s + (0.733 − 0.680i)5-s + (0.826 + 0.563i)9-s + (−0.826 + 0.563i)11-s + (0.900 + 0.433i)13-s + (0.900 − 0.433i)15-s + (−0.365 − 0.930i)17-s + (−0.5 − 0.866i)19-s + (−0.365 + 0.930i)23-s + (0.0747 − 0.997i)25-s + (0.623 + 0.781i)27-s + (0.623 − 0.781i)29-s + (−0.5 + 0.866i)31-s + (−0.955 + 0.294i)33-s + (−0.988 − 0.149i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.0320i$
Analytic conductor: \(0.910220\)
Root analytic conductor: \(0.910220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 196,\ (0:\ ),\ 0.999 + 0.0320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.732424488 + 0.02777060298i\)
\(L(\frac12)\) \(\approx\) \(1.732424488 + 0.02777060298i\)
\(L(1)\) \(\approx\) \(1.508972070 + 0.02467341696i\)
\(L(1)\) \(\approx\) \(1.508972070 + 0.02467341696i\)

Euler product

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

−26.5715943333228997981659411273, −26.02798649276274167351215480131, −25.2854415874384094426709761198, −24.28395228038553079647274819634, −23.30755853359621776890058220398, −22.06349993346399882077373242387, −21.12692006516286219379096765496, −20.46148199664679898296690642681, −19.11294986819509564687114214794, −18.50282116159692978101849767868, −17.646595077542820292704913363267, −16.18378312287248929296603739420, −15.101661440183105414275231108413, −14.25651125678490308004940742950, −13.37403909934219939285772407921, −12.6079474684732045225558980456, −10.8014234117558483985358010904, −10.17800093965093214052989603570, −8.77255688617823059654754230000, −8.01878164572123286466070380615, −6.67262542935985013299990521659, −5.74281318659375731494513521347, −3.87557806415638766033196913101, −2.81042700974210394754615691398, −1.70250057470901795184088437725, 1.67041169168228659086394699354, 2.76996937831660319016302819831, 4.2979225978528248315334042638, 5.253188493818295676988010125606, 6.80309493159017980418184634372, 8.09402984246077900748383905697, 9.05905743881936382376056740868, 9.790910004599682345409941026548, 10.9639099843500238330350297446, 12.55547884127274384662068262570, 13.49005367484443159185305893929, 14.0481402278139738588578337912, 15.54739008185177842675592538669, 16.02093940448171183379543612477, 17.451269777255349375957752729894, 18.33116114443682589755732351308, 19.533576718833312880055676973282, 20.49832624262630071544178293986, 21.08715909201846149452383870316, 21.90406399004007339076945566595, 23.36910413274175760674744357383, 24.30309998394456407180762573721, 25.390593512102544863607787440776, 25.78656206239941371417350968726, 26.83745212775204394101803244836

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