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Properties

Label 2-224-32.29-c1-0-7
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: $\chi_{224} (29, \cdot )$
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

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

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