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Properties

Label 2-224-32.13-c1-0-10
Degree $2$
Conductor $224$
Sign $0.548 - 0.836i$
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  + (−0.539 + 1.30i)2-s + (−0.149 + 0.360i)3-s + (−1.41 − 1.40i)4-s + (2.11 − 0.876i)5-s + (−0.390 − 0.389i)6-s + (0.707 − 0.707i)7-s + (2.60 − 1.09i)8-s + (2.01 + 2.01i)9-s + (0.00537 + 3.23i)10-s + (−0.619 − 1.49i)11-s + (0.719 − 0.300i)12-s + (2.00 + 0.829i)13-s + (0.543 + 1.30i)14-s + 0.892i·15-s + (0.0265 + 3.99i)16-s + 2.15i·17-s + ⋯
L(s)  = 1  + (−0.381 + 0.924i)2-s + (−0.0861 + 0.208i)3-s + (−0.709 − 0.704i)4-s + (0.945 − 0.391i)5-s + (−0.159 − 0.158i)6-s + (0.267 − 0.267i)7-s + (0.921 − 0.387i)8-s + (0.671 + 0.671i)9-s + (0.00169 + 1.02i)10-s + (−0.186 − 0.450i)11-s + (0.207 − 0.0868i)12-s + (0.555 + 0.230i)13-s + (0.145 + 0.348i)14-s + 0.230i·15-s + (0.00663 + 0.999i)16-s + 0.522i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 - 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.548 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.548 - 0.836i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.548 - 0.836i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00661 + 0.543429i\)
\(L(\frac12)\) \(\approx\) \(1.00661 + 0.543429i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.539 - 1.30i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (0.149 - 0.360i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (-2.11 + 0.876i)T + (3.53 - 3.53i)T^{2} \)
11 \( 1 + (0.619 + 1.49i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-2.00 - 0.829i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 2.15iT - 17T^{2} \)
19 \( 1 + (-1.09 - 0.453i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (1.44 + 1.44i)T + 23iT^{2} \)
29 \( 1 + (0.483 - 1.16i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 0.217T + 31T^{2} \)
37 \( 1 + (-9.82 + 4.06i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (4.75 + 4.75i)T + 41iT^{2} \)
43 \( 1 + (1.88 + 4.55i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 6.77iT - 47T^{2} \)
53 \( 1 + (2.22 + 5.37i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (6.35 - 2.63i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (1.20 - 2.90i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-5.29 + 12.7i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (7.30 - 7.30i)T - 71iT^{2} \)
73 \( 1 + (9.12 + 9.12i)T + 73iT^{2} \)
79 \( 1 - 5.17iT - 79T^{2} \)
83 \( 1 + (15.2 + 6.29i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (9.77 - 9.77i)T - 89iT^{2} \)
97 \( 1 + 12.5T + 97T^{2} \)
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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

−12.80629526840971724977687640398, −11.05890701979596525204847750229, −10.26018046201443735966098569740, −9.420297283834945834971373608728, −8.425230778718341434075129686948, −7.45383361524184311350374965287, −6.18601899253058517847373247085, −5.33687160352878086027549000896, −4.20541938369134733408710340579, −1.60056577421926252355573147393, 1.52917719245709547359565113296, 2.90375194813822476755865130640, 4.44164614841454479882458164957, 5.89132953206478689195814449567, 7.16115690482810726282917003991, 8.341752092609066090678925535953, 9.634096405998284376860499310681, 9.939959015562660156169598837960, 11.17121777899031712003885337694, 11.98566440572076954420861961267

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