Properties

Label 2-224-224.109-c1-0-29
Degree $2$
Conductor $224$
Sign $0.221 - 0.975i$
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  + (−0.760 − 1.19i)2-s + (−1.44 − 1.87i)3-s + (−0.842 + 1.81i)4-s + (−2.75 − 2.11i)5-s + (−1.14 + 3.14i)6-s + (−1.14 + 2.38i)7-s + (2.80 − 0.376i)8-s + (−0.673 + 2.51i)9-s + (−0.424 + 4.89i)10-s + (2.68 + 0.353i)11-s + (4.61 − 1.03i)12-s + (1.84 + 0.762i)13-s + (3.71 − 0.450i)14-s + 8.23i·15-s + (−2.58 − 3.05i)16-s + (−6.86 − 3.96i)17-s + ⋯
L(s)  = 1  + (−0.537 − 0.842i)2-s + (−0.831 − 1.08i)3-s + (−0.421 + 0.907i)4-s + (−1.23 − 0.946i)5-s + (−0.466 + 1.28i)6-s + (−0.432 + 0.901i)7-s + (0.991 − 0.132i)8-s + (−0.224 + 0.837i)9-s + (−0.134 + 1.54i)10-s + (0.808 + 0.106i)11-s + (1.33 − 0.297i)12-s + (0.510 + 0.211i)13-s + (0.992 − 0.120i)14-s + 2.12i·15-s + (−0.645 − 0.763i)16-s + (−1.66 − 0.961i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.221 - 0.975i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.221 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0265035 + 0.0211645i\)
\(L(\frac12)\) \(\approx\) \(0.0265035 + 0.0211645i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.760 + 1.19i)T \)
7 \( 1 + (1.14 - 2.38i)T \)
good3 \( 1 + (1.44 + 1.87i)T + (-0.776 + 2.89i)T^{2} \)
5 \( 1 + (2.75 + 2.11i)T + (1.29 + 4.82i)T^{2} \)
11 \( 1 + (-2.68 - 0.353i)T + (10.6 + 2.84i)T^{2} \)
13 \( 1 + (-1.84 - 0.762i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + (6.86 + 3.96i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.226 - 1.71i)T + (-18.3 + 4.91i)T^{2} \)
23 \( 1 + (1.20 - 4.48i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.0946 + 0.228i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (4.72 - 8.18i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.74 + 2.87i)T + (9.57 + 35.7i)T^{2} \)
41 \( 1 + (-0.823 - 0.823i)T + 41iT^{2} \)
43 \( 1 + (2.05 + 4.95i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + (-0.844 + 0.487i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.52 + 0.990i)T + (51.1 + 13.7i)T^{2} \)
59 \( 1 + (-0.187 + 1.42i)T + (-56.9 - 15.2i)T^{2} \)
61 \( 1 + (5.51 - 0.726i)T + (58.9 - 15.7i)T^{2} \)
67 \( 1 + (7.06 + 9.21i)T + (-17.3 + 64.7i)T^{2} \)
71 \( 1 + (0.823 - 0.823i)T - 71iT^{2} \)
73 \( 1 + (-13.8 + 3.71i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.377 - 0.218i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.69 + 3.60i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-6.94 - 1.86i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + 9.00T + 97T^{2} \)
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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

−11.70423133787792116718803240858, −11.06558042764507189686024713837, −9.229556354749216474007693032802, −8.746471170050585795531813242233, −7.55169036320711571298057421335, −6.59546532213044111194054333920, −5.01594269883523523351171538720, −3.66224785266056936060246063596, −1.65499173764098549855411914623, −0.03792322097286779950169232852, 3.88693379664477069034288943305, 4.43251235369532650792577931579, 6.19260757077659577481342681916, 6.81496428588748003815594255315, 7.966249867483070416087030513145, 9.152486892702323039290414352897, 10.31055312593267878495542908623, 10.92006625931103672776772780238, 11.40071030141796727711028405712, 13.12021286039679121338819837639

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