Properties

Label 2-224-56.37-c1-0-1
Degree $2$
Conductor $224$
Sign $0.299 - 0.954i$
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  + (−1.36 + 0.787i)3-s + (0.476 + 0.274i)5-s + (2.60 + 0.447i)7-s + (−0.258 + 0.447i)9-s + (−2.07 + 1.19i)11-s + 3.96i·13-s − 0.866·15-s + (2.10 + 3.65i)17-s + (5.75 + 3.32i)19-s + (−3.91 + 1.44i)21-s + (−1.17 + 2.03i)23-s + (−2.34 − 4.06i)25-s − 5.54i·27-s − 8.21i·29-s + (0.433 + 0.750i)31-s + ⋯
L(s)  = 1  + (−0.787 + 0.454i)3-s + (0.212 + 0.122i)5-s + (0.985 + 0.169i)7-s + (−0.0862 + 0.149i)9-s + (−0.624 + 0.360i)11-s + 1.10i·13-s − 0.223·15-s + (0.511 + 0.885i)17-s + (1.31 + 0.761i)19-s + (−0.853 + 0.314i)21-s + (−0.244 + 0.424i)23-s + (−0.469 − 0.813i)25-s − 1.06i·27-s − 1.52i·29-s + (0.0777 + 0.134i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.299 - 0.954i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.299 - 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.807965 + 0.593485i\)
\(L(\frac12)\) \(\approx\) \(0.807965 + 0.593485i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-2.60 - 0.447i)T \)
good3 \( 1 + (1.36 - 0.787i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.476 - 0.274i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.07 - 1.19i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.96iT - 13T^{2} \)
17 \( 1 + (-2.10 - 3.65i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.75 - 3.32i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.17 - 2.03i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.21iT - 29T^{2} \)
31 \( 1 + (-0.433 - 0.750i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.229 - 0.132i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.24T + 41T^{2} \)
43 \( 1 + 5.35iT - 43T^{2} \)
47 \( 1 + (-1.29 + 2.25i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.36 + 5.40i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.26 - 1.88i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.18 - 3.56i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.31 + 1.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.76T + 71T^{2} \)
73 \( 1 + (2.33 + 4.03i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.308 + 0.533i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.09iT - 83T^{2} \)
89 \( 1 + (-3.19 + 5.53i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
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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

−11.88005777134619101477308417613, −11.73464943478457400692784745387, −10.45515278626296344780350567583, −9.886980059083760942884168423599, −8.414835997009059233102767897026, −7.52609220771178894104623131004, −5.99688122282491416004114052683, −5.21421840380751646515813043316, −4.10306270203651159561723848876, −2.01710376048407178105703777968, 1.00603095853186966520482077326, 3.09044302105114976658539421255, 5.12270071083295227513030617244, 5.56242162067925312582556354663, 7.06885921953238991620137557111, 7.892744996491355814738290015887, 9.100028361806781398305001666386, 10.35684139718886961973014861964, 11.26075221045713715634650093169, 11.91419137982437460372532976595

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