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Properties

Label 2-224-56.3-c1-0-1
Degree $2$
Conductor $224$
Sign $0.824 + 0.565i$
Analytic cond. $1.78864$
Root an. cond. $1.33740$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.18 − 0.686i)3-s + (0.345 + 0.597i)5-s + (2.63 + 0.222i)7-s + (−0.557 − 0.966i)9-s + (1.63 − 2.82i)11-s + 5.27·13-s − 0.947i·15-s + (−2.20 − 1.27i)17-s + (0.484 − 0.279i)19-s + (−2.98 − 2.07i)21-s + (2.50 − 1.44i)23-s + (2.26 − 3.91i)25-s + 5.64i·27-s − 0.444i·29-s + (−4.45 + 7.71i)31-s + ⋯
L(s)  = 1  + (−0.686 − 0.396i)3-s + (0.154 + 0.267i)5-s + (0.996 + 0.0840i)7-s + (−0.185 − 0.322i)9-s + (0.491 − 0.851i)11-s + 1.46·13-s − 0.244i·15-s + (−0.534 − 0.308i)17-s + (0.111 − 0.0642i)19-s + (−0.650 − 0.452i)21-s + (0.522 − 0.301i)23-s + (0.452 − 0.783i)25-s + 1.08i·27-s − 0.0825i·29-s + (−0.799 + 1.38i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.824 + 0.565i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.824 + 0.565i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07912 - 0.334656i\)
\(L(\frac12)\) \(\approx\) \(1.07912 - 0.334656i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-2.63 - 0.222i)T \)
good3 \( 1 + (1.18 + 0.686i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.345 - 0.597i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.63 + 2.82i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.27T + 13T^{2} \)
17 \( 1 + (2.20 + 1.27i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.484 + 0.279i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.50 + 1.44i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.444iT - 29T^{2} \)
31 \( 1 + (4.45 - 7.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.00 - 3.46i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 9.76iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (2.20 + 3.81i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.17 + 4.71i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (8.59 + 4.96i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.23 - 9.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.45 + 2.51i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.29iT - 71T^{2} \)
73 \( 1 + (5.28 + 3.05i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.01 + 2.89i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.83iT - 83T^{2} \)
89 \( 1 + (-1.5 + 0.866i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.42iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.91512230254354141775685966496, −11.23912174697751877792044414029, −10.66268298415281464482322908427, −8.977758998631219437355660168777, −8.349355155470935490950910274293, −6.81433071241681710753875949284, −6.11833166406471587875567237431, −4.95028358443376226661601507317, −3.35055767410698625646299740563, −1.28233500450643950769275025321, 1.71565955527870246854056685864, 3.98105212273386391673769879490, 5.02521687769114316873686548696, 5.95483381892363880126132287818, 7.32309131416298131248198924608, 8.492218820577851254705827509311, 9.405443677855213326546324307269, 10.91093929360230159269360771119, 11.02161747637537463065705652416, 12.18804097880267786029076837020

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