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Properties

Label 2-224-56.51-c2-0-3
Degree $2$
Conductor $224$
Sign $0.993 + 0.114i$
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  + (−2.25 − 3.90i)3-s + (6.07 + 3.50i)5-s + (2.51 + 6.53i)7-s + (−5.64 + 9.78i)9-s + (7.90 + 13.6i)11-s − 2.90i·13-s − 31.5i·15-s + (1.65 + 2.86i)17-s + (8.10 − 14.0i)19-s + (19.8 − 24.5i)21-s + (16.4 + 9.51i)23-s + (12.0 + 20.9i)25-s + 10.3·27-s − 21.1i·29-s + (23.6 − 13.6i)31-s + ⋯
L(s)  = 1  + (−0.750 − 1.30i)3-s + (1.21 + 0.701i)5-s + (0.358 + 0.933i)7-s + (−0.627 + 1.08i)9-s + (0.718 + 1.24i)11-s − 0.223i·13-s − 2.10i·15-s + (0.0973 + 0.168i)17-s + (0.426 − 0.739i)19-s + (0.944 − 1.16i)21-s + (0.716 + 0.413i)23-s + (0.482 + 0.836i)25-s + 0.383·27-s − 0.728i·29-s + (0.764 − 0.441i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.993 + 0.114i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ 0.993 + 0.114i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.57185 - 0.0899177i\)
\(L(\frac12)\) \(\approx\) \(1.57185 - 0.0899177i\)
\(L(2)\) 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 \( 1 + (-2.51 - 6.53i)T \)
good3 \( 1 + (2.25 + 3.90i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-6.07 - 3.50i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-7.90 - 13.6i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 2.90iT - 169T^{2} \)
17 \( 1 + (-1.65 - 2.86i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-8.10 + 14.0i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-16.4 - 9.51i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 21.1iT - 841T^{2} \)
31 \( 1 + (-23.6 + 13.6i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-15.2 - 8.82i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 1.13T + 1.68e3T^{2} \)
43 \( 1 + 50.2T + 1.84e3T^{2} \)
47 \( 1 + (0.657 + 0.379i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (38.9 - 22.4i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-1.19 - 2.06i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-86.1 - 49.7i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (33.2 + 57.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 44.4iT - 5.04e3T^{2} \)
73 \( 1 + (0.859 + 1.48i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (62.9 + 36.3i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 102.T + 6.88e3T^{2} \)
89 \( 1 + (-30.6 + 53.1i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 102.T + 9.40e3T^{2} \)
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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

−11.97199750471203306206402475136, −11.37954415604749198623301269571, −10.07012754186156238792433173377, −9.187843040029486725455753355095, −7.69962063654012795795293217237, −6.70537159451496505113885386882, −6.08448728395645876177024825579, −5.03276848306059992434617756683, −2.53177490704446529448379826994, −1.54017641309873042957105057455, 1.13124862105895924576446937378, 3.59449405064223387114099222430, 4.78472482190215159387527049071, 5.56449557709866830801023571929, 6.60112475984977860008639289092, 8.418853129073263347402294660000, 9.395308096435366997182906195622, 10.11385908574992653839015409250, 10.92856989862466816364452233223, 11.73548703935413207893914242652

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