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Properties

Label 2-224-7.2-c3-0-12
Degree $2$
Conductor $224$
Sign $0.687 + 0.726i$
Analytic cond. $13.2164$
Root an. cond. $3.63544$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.78 − 4.82i)3-s + (2.27 − 3.94i)5-s + (3.15 + 18.2i)7-s + (−2.02 + 3.49i)9-s + (32.8 + 56.9i)11-s + 73.6·13-s − 25.3·15-s + (−65.5 − 113. i)17-s + (13.2 − 22.9i)19-s + (79.2 − 66.0i)21-s + (41.0 − 71.1i)23-s + (52.1 + 90.2i)25-s − 127.·27-s + 4.65·29-s + (−40.0 − 69.3i)31-s + ⋯
L(s)  = 1  + (−0.536 − 0.928i)3-s + (0.203 − 0.352i)5-s + (0.170 + 0.985i)7-s + (−0.0748 + 0.129i)9-s + (0.901 + 1.56i)11-s + 1.57·13-s − 0.436·15-s + (−0.934 − 1.61i)17-s + (0.159 − 0.276i)19-s + (0.823 − 0.686i)21-s + (0.372 − 0.644i)23-s + (0.417 + 0.722i)25-s − 0.911·27-s + 0.0297·29-s + (−0.231 − 0.401i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.687 + 0.726i$
Analytic conductor: \(13.2164\)
Root analytic conductor: \(3.63544\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :3/2),\ 0.687 + 0.726i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.57405 - 0.677210i\)
\(L(\frac12)\) \(\approx\) \(1.57405 - 0.677210i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.15 - 18.2i)T \)
good3 \( 1 + (2.78 + 4.82i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-2.27 + 3.94i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-32.8 - 56.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 73.6T + 2.19e3T^{2} \)
17 \( 1 + (65.5 + 113. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-13.2 + 22.9i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-41.0 + 71.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 4.65T + 2.43e4T^{2} \)
31 \( 1 + (40.0 + 69.3i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-133. + 231. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 328.T + 6.89e4T^{2} \)
43 \( 1 - 289.T + 7.95e4T^{2} \)
47 \( 1 + (-59.6 + 103. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-75.1 - 130. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-211. - 366. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-251. + 435. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-195. - 337. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 803.T + 3.57e5T^{2} \)
73 \( 1 + (-184. - 319. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-276. + 478. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 62.3T + 5.71e5T^{2} \)
89 \( 1 + (237. - 411. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 11.8T + 9.12e5T^{2} \)
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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

−11.75173655430929045191599695126, −11.12818977700282630199234785024, −9.356998889513233664248733603898, −8.975553716868013778758929093767, −7.38906597435435209825579361171, −6.60954468400735213417979154800, −5.60553067979666563291473479982, −4.33374359270074401303705803408, −2.28021684232974657740618085612, −1.02149478716968634065223456058, 1.16369431446551463813621116329, 3.58703909466870277535015874588, 4.23141337222530516751005647930, 5.84865902880684659090010380507, 6.50179031739199226646892384509, 8.127505499415092145939203235497, 9.039334405843071927084245332816, 10.35751643034342367353200298100, 10.94441222579500325484176099809, 11.35220592585180650278123456676

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