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Properties

Label 2-224-7.4-c1-0-0
Degree $2$
Conductor $224$
Sign $-0.832 - 0.553i$
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  + (−1.32 + 2.29i)3-s + (1.5 + 2.59i)5-s − 2.64·7-s + (−2 − 3.46i)9-s + (1.32 − 2.29i)11-s − 4·13-s − 7.93·15-s + (−0.5 + 0.866i)17-s + (3.96 + 6.87i)19-s + (3.50 − 6.06i)21-s + (1.32 + 2.29i)23-s + (−2 + 3.46i)25-s + 2.64·27-s − 4·29-s + (1.32 − 2.29i)31-s + ⋯
L(s)  = 1  + (−0.763 + 1.32i)3-s + (0.670 + 1.16i)5-s − 0.999·7-s + (−0.666 − 1.15i)9-s + (0.398 − 0.690i)11-s − 1.10·13-s − 2.04·15-s + (−0.121 + 0.210i)17-s + (0.910 + 1.57i)19-s + (0.763 − 1.32i)21-s + (0.275 + 0.477i)23-s + (−0.400 + 0.692i)25-s + 0.509·27-s − 0.742·29-s + (0.237 − 0.411i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.832 - 0.553i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ -0.832 - 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.240222 + 0.794876i\)
\(L(\frac12)\) \(\approx\) \(0.240222 + 0.794876i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.64T \)
good3 \( 1 + (1.32 - 2.29i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.32 + 2.29i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.96 - 6.87i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.32 - 2.29i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (-1.32 + 2.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 + (1.32 + 2.29i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.5 - 6.06i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.32 - 2.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.32 - 2.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-4.5 + 7.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.32 - 2.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−12.43051122780689631768920768148, −11.39866300715551577242685085300, −10.56061436602204266487540962884, −9.838785668080396352477306989906, −9.371314119906094528772280788488, −7.43887328406616401627480960166, −6.15756708995599841641975974475, −5.62737362992219887718834288985, −4.00221241845526800761500765871, −2.93437558179583465315254337727, 0.75769452215662175966794897536, 2.37326236918796427722484503952, 4.75412054384598175316657595680, 5.72395930975516652537888083230, 6.80738120356145628330568814239, 7.48772715194962209275591511857, 9.157874530796429338673688796185, 9.625179439087469050500614364841, 11.19318640966380535600237616067, 12.27063203644167612815678198091

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