Properties

Label 2-224-32.13-c1-0-19
Degree $2$
Conductor $224$
Sign $0.334 + 0.942i$
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  + (−1.04 + 0.949i)2-s + (1.17 − 2.83i)3-s + (0.198 − 1.99i)4-s + (0.671 − 0.278i)5-s + (1.45 + 4.08i)6-s + (0.707 − 0.707i)7-s + (1.68 + 2.27i)8-s + (−4.52 − 4.52i)9-s + (−0.440 + 0.928i)10-s + (1.28 + 3.09i)11-s + (−5.40 − 2.89i)12-s + (−1.28 − 0.531i)13-s + (−0.0702 + 1.41i)14-s − 2.22i·15-s + (−3.92 − 0.789i)16-s − 7.04i·17-s + ⋯
L(s)  = 1  + (−0.741 + 0.671i)2-s + (0.677 − 1.63i)3-s + (0.0991 − 0.995i)4-s + (0.300 − 0.124i)5-s + (0.595 + 1.66i)6-s + (0.267 − 0.267i)7-s + (0.594 + 0.804i)8-s + (−1.50 − 1.50i)9-s + (−0.139 + 0.293i)10-s + (0.386 + 0.932i)11-s + (−1.55 − 0.836i)12-s + (−0.355 − 0.147i)13-s + (−0.0187 + 0.377i)14-s − 0.575i·15-s + (−0.980 − 0.197i)16-s − 1.70i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.334 + 0.942i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.334 + 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.887027 - 0.626653i\)
\(L(\frac12)\) \(\approx\) \(0.887027 - 0.626653i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.04 - 0.949i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-1.17 + 2.83i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (-0.671 + 0.278i)T + (3.53 - 3.53i)T^{2} \)
11 \( 1 + (-1.28 - 3.09i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (1.28 + 0.531i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 7.04iT - 17T^{2} \)
19 \( 1 + (3.86 + 1.60i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-6.68 - 6.68i)T + 23iT^{2} \)
29 \( 1 + (-0.00762 + 0.0184i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 6.26T + 31T^{2} \)
37 \( 1 + (-1.15 + 0.478i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-2.46 - 2.46i)T + 41iT^{2} \)
43 \( 1 + (0.319 + 0.772i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 - 3.71iT - 47T^{2} \)
53 \( 1 + (-2.37 - 5.74i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (6.83 - 2.83i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-0.476 + 1.15i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (0.0478 - 0.115i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-6.58 + 6.58i)T - 71iT^{2} \)
73 \( 1 + (1.63 + 1.63i)T + 73iT^{2} \)
79 \( 1 - 12.5iT - 79T^{2} \)
83 \( 1 + (-15.2 - 6.32i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-8.92 + 8.92i)T - 89iT^{2} \)
97 \( 1 - 1.31T + 97T^{2} \)
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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

−12.13356747855029456035192868568, −11.19418131903357595582656324157, −9.589385944555864894032816667436, −9.033227552355471668328206910769, −7.74353417932513048133232341878, −7.27964627729393938750610335611, −6.44625796183601660583859630059, −5.02019533352792799303759512065, −2.50067531799584271008652288143, −1.20180342046771399347486937426, 2.40189687525531083457586442821, 3.62912915614460346248047257477, 4.58256282682581759582542788530, 6.27425351225097884366090346321, 8.308060893093775868936219976638, 8.588955933919156679550484009241, 9.606473258825529163887168880354, 10.53277770086821623912441899151, 10.89538293710708806621833542756, 12.17043277064933591266210674144

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