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Properties

Label 2-224-28.3-c1-0-1
Degree $2$
Conductor $224$
Sign $-0.996 - 0.0805i$
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  + (−1.60 + 2.77i)3-s + (−1.00 + 0.579i)5-s + (1.44 + 2.21i)7-s + (−3.63 − 6.28i)9-s + (−3.93 − 2.27i)11-s + 2.08i·13-s − 3.71i·15-s + (−0.301 − 0.174i)17-s + (0.156 + 0.270i)19-s + (−8.46 + 0.460i)21-s + (−4.08 + 2.35i)23-s + (−1.82 + 3.16i)25-s + 13.6·27-s + 7.26·29-s + (−1.13 + 1.96i)31-s + ⋯
L(s)  = 1  + (−0.924 + 1.60i)3-s + (−0.448 + 0.259i)5-s + (0.546 + 0.837i)7-s + (−1.21 − 2.09i)9-s + (−1.18 − 0.684i)11-s + 0.577i·13-s − 0.958i·15-s + (−0.0731 − 0.0422i)17-s + (0.0358 + 0.0620i)19-s + (−1.84 + 0.100i)21-s + (−0.852 + 0.492i)23-s + (−0.365 + 0.633i)25-s + 2.62·27-s + 1.34·29-s + (−0.203 + 0.352i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.996 - 0.0805i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ -0.996 - 0.0805i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0237633 + 0.588959i\)
\(L(\frac12)\) \(\approx\) \(0.0237633 + 0.588959i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.44 - 2.21i)T \)
good3 \( 1 + (1.60 - 2.77i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.00 - 0.579i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.93 + 2.27i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.08iT - 13T^{2} \)
17 \( 1 + (0.301 + 0.174i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.156 - 0.270i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.08 - 2.35i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.26T + 29T^{2} \)
31 \( 1 + (1.13 - 1.96i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.63 - 6.29i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.08iT - 41T^{2} \)
43 \( 1 + 1.43iT - 43T^{2} \)
47 \( 1 + (-4.02 - 6.97i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.805 + 1.39i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.478 + 0.829i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.88 - 5.70i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.592 + 0.342i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.57iT - 71T^{2} \)
73 \( 1 + (-7.58 - 4.38i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-13.6 + 7.90i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 + (7.11 - 4.10i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.0iT - 97T^{2} \)
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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

−12.23373831790360909374341123890, −11.51207223068673475365942557849, −10.89072485711643343102063266736, −10.00520986023260379987927053802, −9.002962078750872723166850629663, −7.967892286621497067860055698858, −6.18902800295395615617974586273, −5.32486884936053109128013726545, −4.41804384134068620305011706288, −3.07558130716892031297653704692, 0.53588196844997373337365547622, 2.22913116438187334526880222024, 4.55088722061057463885011072699, 5.64123308293591153567908641845, 6.83867533210904925945694189639, 7.76920326157522998830706671937, 8.155013738190182859131194279747, 10.27423983390063087041172042617, 10.96045219664593237811347906078, 12.03397598346698331912863180697

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