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Properties

Label 2-224-56.3-c1-0-2
Degree $2$
Conductor $224$
Sign $0.877 - 0.478i$
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  + (2.27 + 1.31i)3-s + (−1.03 − 1.80i)5-s + (1.25 + 2.33i)7-s + (1.94 + 3.36i)9-s + (0.669 − 1.16i)11-s + 2.50·13-s − 5.45i·15-s + (−2.78 − 1.60i)17-s + (−3.55 + 2.05i)19-s + (−0.212 + 6.93i)21-s + (−5.54 + 3.20i)23-s + (0.339 − 0.588i)25-s + 2.32i·27-s − 4.66i·29-s + (2.21 − 3.84i)31-s + ⋯
L(s)  = 1  + (1.31 + 0.757i)3-s + (−0.464 − 0.805i)5-s + (0.473 + 0.880i)7-s + (0.647 + 1.12i)9-s + (0.201 − 0.349i)11-s + 0.694·13-s − 1.40i·15-s + (−0.674 − 0.389i)17-s + (−0.815 + 0.470i)19-s + (−0.0464 + 1.51i)21-s + (−1.15 + 0.668i)23-s + (0.0679 − 0.117i)25-s + 0.446i·27-s − 0.865i·29-s + (0.398 − 0.690i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.877 - 0.478i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.877 - 0.478i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66932 + 0.425571i\)
\(L(\frac12)\) \(\approx\) \(1.66932 + 0.425571i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.25 - 2.33i)T \)
good3 \( 1 + (-2.27 - 1.31i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.03 + 1.80i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.669 + 1.16i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.50T + 13T^{2} \)
17 \( 1 + (2.78 + 1.60i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.55 - 2.05i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.54 - 3.20i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.66iT - 29T^{2} \)
31 \( 1 + (-2.21 + 3.84i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.50 - 3.17i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.55iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (0.565 + 0.980i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.43 - 4.29i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.29 + 3.63i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.57 + 4.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.93 - 6.81i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.29iT - 71T^{2} \)
73 \( 1 + (-0.480 - 0.277i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.26 + 3.04i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.503iT - 83T^{2} \)
89 \( 1 + (-1.5 + 0.866i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 17.2iT - 97T^{2} \)
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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

−12.29118266430117306224828243283, −11.44487588988726338307468698117, −10.16551013581731156151949703993, −9.064950090739253687525584113583, −8.557750029331530162878359971948, −7.906026377947713276325543600482, −6.01379262120526397472464659484, −4.60506128559483100061189975950, −3.72187894487126214740263171688, −2.19957900587895392694730915448, 1.83556472642489875635684806792, 3.26404006395518590096881464735, 4.32436221695883982973341295774, 6.55919998051828427852644076225, 7.22491264448357714099889796714, 8.171563613500282618598613347359, 8.885814753743470497865526966227, 10.38058333013966707927614090269, 11.08495576623424225785972769119, 12.37371442123061375468482508182

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