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Properties

Label 1-14e2-196.151-r1-0-0
Degree $1$
Conductor $196$
Sign $-0.424 - 0.905i$
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  + (−0.0747 − 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

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

Invariants

Degree: \(1\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.424 - 0.905i$
Analytic conductor: \(21.0631\)
Root analytic conductor: \(21.0631\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 196,\ (1:\ ),\ -0.424 - 0.905i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.144165933 - 1.799514913i\)
\(L(\frac12)\) \(\approx\) \(1.144165933 - 1.799514913i\)
\(L(1)\) \(\approx\) \(1.118785383 - 0.6564273851i\)
\(L(1)\) \(\approx\) \(1.118785383 - 0.6564273851i\)

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

−27.03291705706978524724452362494, −26.03673559265656132444535561001, −25.51284956183013439948549884718, −24.27956235925706370560607200866, −22.87279392843152535417330229865, −22.30162725977791414022362772385, −21.29513706901895724210689330651, −20.751891993028544320947821525327, −19.39480929050719727421511912566, −18.38204142017155929090968744803, −17.161539889300036878638516966737, −16.56907867745944470337490381509, −15.36244395522496985708722544252, −14.28122535106552765387601866993, −13.82156044676035430077928372236, −12.01438941079408619238312231001, −11.13975177542918727076945106392, −9.942467583036685311380195576140, −9.449294531447597182768844207, −8.098048379849776937730137653621, −6.37905954018625129223059991792, −5.72554068851118908041107703211, −4.18823754629617860368555972496, −3.22455300370286750981432905180, −1.59749306007858039469304630001, 0.79521349539781435544600275784, 1.78450628076853519649819856949, 3.26604665520941521758101156949, 5.12042698939608653707812124912, 6.046695494176732306656418171736, 7.08544432147768077402958332835, 8.363911925504368894262874460906, 9.26315838712043775635523289036, 10.57331459738187452264557276730, 11.95152880171762289531238239016, 12.63714781556668352277517746825, 13.7404313161017888909339320735, 14.321965947927204688252988185159, 15.95096408245989203571037787450, 17.06958006039089499462010563563, 17.755104707070710875111922293305, 18.60153072645719724665392570421, 19.87283212688474390995615234598, 20.46846823996178613497374814684, 21.79880902931731697013350422182, 22.715693848834131651561381658267, 23.75814978494422386575096045350, 24.70320515320264757254414784565, 25.26790441650116930550686737271, 26.09561265849020310233814006001

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