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Properties

Label 2-224-56.11-c2-0-3
Degree $2$
Conductor $224$
Sign $-0.727 - 0.685i$
Analytic cond. $6.10355$
Root an. cond. $2.47053$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.99 + 3.44i)3-s + (1.63 − 0.941i)5-s + (5.14 + 4.74i)7-s + (−3.42 − 5.93i)9-s + (−3.93 + 6.82i)11-s + 11.4i·13-s + 7.49i·15-s + (1.44 − 2.51i)17-s + (−15.0 − 26.0i)19-s + (−26.6 + 8.30i)21-s + (−33.3 + 19.2i)23-s + (−10.7 + 18.5i)25-s − 8.56·27-s + 27.8i·29-s + (19.4 + 11.2i)31-s + ⋯
L(s)  = 1  + (−0.663 + 1.14i)3-s + (0.326 − 0.188i)5-s + (0.735 + 0.677i)7-s + (−0.380 − 0.659i)9-s + (−0.358 + 0.620i)11-s + 0.883i·13-s + 0.499i·15-s + (0.0852 − 0.147i)17-s + (−0.790 − 1.36i)19-s + (−1.26 + 0.395i)21-s + (−1.45 + 0.838i)23-s + (−0.429 + 0.743i)25-s − 0.317·27-s + 0.961i·29-s + (0.628 + 0.362i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 - 0.685i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.727 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.727 - 0.685i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ -0.727 - 0.685i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.413712 + 1.04251i\)
\(L(\frac12)\) \(\approx\) \(0.413712 + 1.04251i\)
\(L(2)\) 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 \( 1 + (-5.14 - 4.74i)T \)
good3 \( 1 + (1.99 - 3.44i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (-1.63 + 0.941i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (3.93 - 6.82i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 11.4iT - 169T^{2} \)
17 \( 1 + (-1.44 + 2.51i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (15.0 + 26.0i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (33.3 - 19.2i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 27.8iT - 841T^{2} \)
31 \( 1 + (-19.4 - 11.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-39.4 + 22.7i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 40.6T + 1.68e3T^{2} \)
43 \( 1 - 47.2T + 1.84e3T^{2} \)
47 \( 1 + (-71.5 + 41.2i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-23.2 - 13.4i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (5.20 - 9.01i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-19.1 + 11.0i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (29.6 - 51.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 38.2iT - 5.04e3T^{2} \)
73 \( 1 + (6.98 - 12.1i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-44.3 + 25.5i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 89.4T + 6.88e3T^{2} \)
89 \( 1 + (52.6 + 91.1i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 55.3T + 9.40e3T^{2} \)
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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

−12.06584021180470786240424809219, −11.37123434322681274845072305296, −10.49634959556840202277519293895, −9.527563865571843782226841473974, −8.794071121528801267550127568962, −7.35144000325860559925199777600, −5.87065942547674838441763570383, −5.00542825763347777826369862417, −4.16828831378454160614711261999, −2.12851134836776166060096886117, 0.66116450474047397058457746148, 2.16828732797048316427962863867, 4.15878366722422013744310196151, 5.81482535228401781793055400305, 6.32560794300330983886956242329, 7.83872452530272777438393483515, 8.104169756253099490434230213828, 10.08396009730295422911987897230, 10.71803073066744275461491375205, 11.82695907540510645289779144220

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