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Properties

Label 2-224-224.195-c1-0-12
Degree $2$
Conductor $224$
Sign $0.678 - 0.734i$
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  + (1.28 + 0.581i)2-s + (1.32 + 1.50i)4-s + (1.87 + 1.87i)7-s + (0.832 + 2.70i)8-s + (−2.12 − 2.12i)9-s + (−0.639 − 1.54i)11-s + (1.32 + 3.5i)14-s + (−0.5 + 3.96i)16-s + (−1.5 − 3.96i)18-s + (0.0740 − 2.36i)22-s + (6.64 − 6.64i)23-s + (−3.53 + 3.53i)25-s + (−0.331 + 5.28i)28-s + (−9.74 − 4.03i)29-s + (−2.95 + 4.82i)32-s + ⋯
L(s)  = 1  + (0.911 + 0.411i)2-s + (0.661 + 0.750i)4-s + (0.707 + 0.707i)7-s + (0.294 + 0.955i)8-s + (−0.707 − 0.707i)9-s + (−0.192 − 0.465i)11-s + (0.353 + 0.935i)14-s + (−0.125 + 0.992i)16-s + (−0.353 − 0.935i)18-s + (0.0157 − 0.503i)22-s + (1.38 − 1.38i)23-s + (−0.707 + 0.707i)25-s + (−0.0626 + 0.998i)28-s + (−1.80 − 0.749i)29-s + (−0.522 + 0.852i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.678 - 0.734i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.678 - 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85228 + 0.810667i\)
\(L(\frac12)\) \(\approx\) \(1.85228 + 0.810667i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 - 0.581i)T \)
7 \( 1 + (-1.87 - 1.87i)T \)
good3 \( 1 + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (3.53 - 3.53i)T^{2} \)
11 \( 1 + (0.639 + 1.54i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-6.64 + 6.64i)T - 23iT^{2} \)
29 \( 1 + (9.74 + 4.03i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (3.67 + 8.86i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 - 41iT^{2} \)
43 \( 1 + (-0.113 - 0.273i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + (-4.79 + 1.98i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (5.02 - 12.1i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-11.3 - 11.3i)T + 71iT^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 - 5.56T + 79T^{2} \)
83 \( 1 + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + 89iT^{2} \)
97 \( 1 - 97T^{2} \)
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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

−12.46064748762069349259945558572, −11.51624260638522569732509760965, −10.96054169126174182670344428102, −9.137005130945523702314601424923, −8.358267356145680147085408127202, −7.20831752116406898405455009605, −5.94542007594034939001434359750, −5.25308633290917300584645364844, −3.79630310957879825085184628091, −2.45827027381107723535448138779, 1.81351784080627547355671871761, 3.39545982717243189919977493041, 4.75331792737552615199742578861, 5.53233975942623609269331259780, 7.02472603792579828043684182817, 7.913736500635687929765602657897, 9.431752153669395666206595198036, 10.62330568738277128261618853092, 11.18421396858747236942336129461, 12.06520824621367846838783653572

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