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Properties

Label 2-224-32.21-c1-0-0
Degree $2$
Conductor $224$
Sign $-0.00334 + 0.999i$
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  + (−0.0564 + 1.41i)2-s + (−2.95 + 1.22i)3-s + (−1.99 − 0.159i)4-s + (−1.21 + 2.92i)5-s + (−1.56 − 4.24i)6-s + (0.707 + 0.707i)7-s + (0.338 − 2.80i)8-s + (5.11 − 5.11i)9-s + (−4.06 − 1.87i)10-s + (−1.89 − 0.786i)11-s + (6.08 − 1.96i)12-s + (1.39 + 3.36i)13-s + (−1.03 + 0.959i)14-s − 10.1i·15-s + (3.94 + 0.636i)16-s − 1.19i·17-s + ⋯
L(s)  = 1  + (−0.0399 + 0.999i)2-s + (−1.70 + 0.706i)3-s + (−0.996 − 0.0797i)4-s + (−0.541 + 1.30i)5-s + (−0.638 − 1.73i)6-s + (0.267 + 0.267i)7-s + (0.119 − 0.992i)8-s + (1.70 − 1.70i)9-s + (−1.28 − 0.593i)10-s + (−0.572 − 0.237i)11-s + (1.75 − 0.568i)12-s + (0.386 + 0.932i)13-s + (−0.277 + 0.256i)14-s − 2.61i·15-s + (0.987 + 0.159i)16-s − 0.289i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00334 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00334 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.00334 + 0.999i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ -0.00334 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.166911 - 0.167470i\)
\(L(\frac12)\) \(\approx\) \(0.166911 - 0.167470i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0564 - 1.41i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (2.95 - 1.22i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (1.21 - 2.92i)T + (-3.53 - 3.53i)T^{2} \)
11 \( 1 + (1.89 + 0.786i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (-1.39 - 3.36i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + 1.19iT - 17T^{2} \)
19 \( 1 + (1.98 + 4.80i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (4.77 - 4.77i)T - 23iT^{2} \)
29 \( 1 + (1.61 - 0.670i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + 1.81T + 31T^{2} \)
37 \( 1 + (-3.48 + 8.41i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (6.87 - 6.87i)T - 41iT^{2} \)
43 \( 1 + (-2.37 - 0.984i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 1.33iT - 47T^{2} \)
53 \( 1 + (5.26 + 2.17i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-3.13 + 7.57i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (4.76 - 1.97i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (9.61 - 3.98i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-6.26 - 6.26i)T + 71iT^{2} \)
73 \( 1 + (-0.436 + 0.436i)T - 73iT^{2} \)
79 \( 1 + 4.79iT - 79T^{2} \)
83 \( 1 + (-2.57 - 6.22i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (8.88 + 8.88i)T + 89iT^{2} \)
97 \( 1 + 5.09T + 97T^{2} \)
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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

−12.92346490665306698374637563978, −11.59441647375925432814992600496, −11.15281485340287628650177520869, −10.21624168022044654539102047777, −9.199995390602436559286392302090, −7.59302362416472301093065939176, −6.69377025115830688279416763038, −5.94233575517654275220500194225, −4.85731637969166190703103483259, −3.77877944309056878755989657876, 0.25850699132329544709744865360, 1.56928604677385177318322933142, 4.19858496082960933180447825556, 5.07565746502223809530356177244, 5.97206552054773086575898280152, 7.76079521395747544583893267796, 8.402477866696972242452328008776, 10.14639647220807290122284239646, 10.71314662401788778612412831084, 11.74902838191480490163512161020

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