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Properties

Label 2-224-32.21-c1-0-21
Degree $2$
Conductor $224$
Sign $-0.315 + 0.948i$
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  + (0.146 − 1.40i)2-s + (2.41 − 0.998i)3-s + (−1.95 − 0.411i)4-s + (0.0734 − 0.177i)5-s + (−1.05 − 3.53i)6-s + (−0.707 − 0.707i)7-s + (−0.864 + 2.69i)8-s + (2.69 − 2.69i)9-s + (−0.238 − 0.129i)10-s + (0.558 + 0.231i)11-s + (−5.12 + 0.962i)12-s + (−1.02 − 2.46i)13-s + (−1.09 + 0.891i)14-s − 0.500i·15-s + (3.66 + 1.60i)16-s + 1.00i·17-s + ⋯
L(s)  = 1  + (0.103 − 0.994i)2-s + (1.39 − 0.576i)3-s + (−0.978 − 0.205i)4-s + (0.0328 − 0.0793i)5-s + (−0.429 − 1.44i)6-s + (−0.267 − 0.267i)7-s + (−0.305 + 0.952i)8-s + (0.897 − 0.897i)9-s + (−0.0754 − 0.0408i)10-s + (0.168 + 0.0697i)11-s + (−1.48 + 0.277i)12-s + (−0.283 − 0.684i)13-s + (−0.293 + 0.238i)14-s − 0.129i·15-s + (0.915 + 0.402i)16-s + 0.242i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.315 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.315 + 0.948i$
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: \((2,\ 224,\ (\ :1/2),\ -0.315 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.998302 - 1.38461i\)
\(L(\frac12)\) \(\approx\) \(0.998302 - 1.38461i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.146 + 1.40i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-2.41 + 0.998i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (-0.0734 + 0.177i)T + (-3.53 - 3.53i)T^{2} \)
11 \( 1 + (-0.558 - 0.231i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (1.02 + 2.46i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 - 1.00iT - 17T^{2} \)
19 \( 1 + (-2.49 - 6.03i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-1.05 + 1.05i)T - 23iT^{2} \)
29 \( 1 + (1.79 - 0.743i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + 6.06T + 31T^{2} \)
37 \( 1 + (1.03 - 2.48i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (-7.82 + 7.82i)T - 41iT^{2} \)
43 \( 1 + (9.68 + 4.01i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 11.0iT - 47T^{2} \)
53 \( 1 + (-5.84 - 2.42i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-3.11 + 7.51i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-5.06 + 2.09i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (8.30 - 3.44i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (6.50 + 6.50i)T + 71iT^{2} \)
73 \( 1 + (-1.02 + 1.02i)T - 73iT^{2} \)
79 \( 1 - 16.1iT - 79T^{2} \)
83 \( 1 + (4.25 + 10.2i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-5.60 - 5.60i)T + 89iT^{2} \)
97 \( 1 + 13.7T + 97T^{2} \)
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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

−12.33888276196574110984128456017, −10.94155399100788122879386585746, −9.927831406695958626403038300961, −9.098139220980619888932394224771, −8.202893210485917889380958188862, −7.27681519568209321326818050445, −5.51427263183955963347812528228, −3.84271249101142542372032305436, −2.96641991557417385621539360436, −1.57724576553883033245273138781, 2.80087490559061043527040173616, 4.00443344109435174299472740813, 5.12988353773111766917962208597, 6.68055127698667940466223647175, 7.59468979289232793177490571846, 8.799244794607224727135550556563, 9.184712456635506957065553037222, 10.09050105414116687347125572303, 11.67992925013999966056883630295, 13.09537216766915543268619641326

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