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Properties

Label 2-224-224.69-c0-0-0
Degree $2$
Conductor $224$
Sign $0.555 - 0.831i$
Analytic cond. $0.111790$
Root an. cond. $0.334350$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.707 − 1.70i)11-s − 1.00i·14-s − 1.00·16-s − 1.00·18-s + (1.70 − 0.707i)22-s + (−1 + i)23-s + (0.707 + 0.707i)25-s + (0.707 − 0.707i)28-s + (−0.707 − 1.70i)29-s + (−0.707 − 0.707i)32-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.707 − 1.70i)11-s − 1.00i·14-s − 1.00·16-s − 1.00·18-s + (1.70 − 0.707i)22-s + (−1 + i)23-s + (0.707 + 0.707i)25-s + (0.707 − 0.707i)28-s + (−0.707 − 1.70i)29-s + (−0.707 − 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.555 - 0.831i$
Analytic conductor: \(0.111790\)
Root analytic conductor: \(0.334350\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :0),\ 0.555 - 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9147878900\)
\(L(\frac12)\) \(\approx\) \(0.9147878900\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (0.707 - 0.707i)T^{2} \)
5 \( 1 + (-0.707 - 0.707i)T^{2} \)
11 \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \)
13 \( 1 + (-0.707 + 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + (1 - i)T - iT^{2} \)
29 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.707 - 0.707i)T^{2} \)
67 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 - T^{2} \)
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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

−13.01896047236179747630980474753, −11.62413417223418457936967062638, −11.08933817524096374040867117362, −9.558664557522519020388867669763, −8.434536957138598386964682300441, −7.58349170248394100224433670765, −6.29893581866913974948321944843, −5.63213365356674983441617051222, −4.05555792793186239881001508243, −3.05489877579825797630944806539, 2.21337394587585197564071831515, 3.55072378156165014876097734764, 4.81090955923048536674171304725, 6.10004080784686032476692554731, 6.88124373243444318155137645877, 8.823579721935180019121213818173, 9.545982489762076035360758359071, 10.44510991769651225499325994251, 11.74811795196556075540511044261, 12.35448965942733587960770380009

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