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Properties

Label 2-224-224.115-c1-0-2
Degree $2$
Conductor $224$
Sign $0.296 - 0.955i$
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  + (−0.324 − 1.37i)2-s + (0.328 + 0.427i)3-s + (−1.78 + 0.894i)4-s + (−1.66 + 2.17i)5-s + (0.482 − 0.590i)6-s + (−2.39 + 1.13i)7-s + (1.81 + 2.17i)8-s + (0.701 − 2.61i)9-s + (3.53 + 1.59i)10-s + (−0.567 + 4.30i)11-s + (−0.969 − 0.471i)12-s + (−1.99 + 4.82i)13-s + (2.33 + 2.92i)14-s − 1.47·15-s + (2.40 − 3.19i)16-s + (0.330 − 0.572i)17-s + ⋯
L(s)  = 1  + (−0.229 − 0.973i)2-s + (0.189 + 0.246i)3-s + (−0.894 + 0.447i)4-s + (−0.746 + 0.973i)5-s + (0.196 − 0.241i)6-s + (−0.903 + 0.428i)7-s + (0.640 + 0.767i)8-s + (0.233 − 0.872i)9-s + (1.11 + 0.503i)10-s + (−0.171 + 1.29i)11-s + (−0.279 − 0.136i)12-s + (−0.554 + 1.33i)13-s + (0.624 + 0.780i)14-s − 0.381·15-s + (0.600 − 0.799i)16-s + (0.0801 − 0.138i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.296 - 0.955i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.296 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.498875 + 0.367672i\)
\(L(\frac12)\) \(\approx\) \(0.498875 + 0.367672i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.324 + 1.37i)T \)
7 \( 1 + (2.39 - 1.13i)T \)
good3 \( 1 + (-0.328 - 0.427i)T + (-0.776 + 2.89i)T^{2} \)
5 \( 1 + (1.66 - 2.17i)T + (-1.29 - 4.82i)T^{2} \)
11 \( 1 + (0.567 - 4.30i)T + (-10.6 - 2.84i)T^{2} \)
13 \( 1 + (1.99 - 4.82i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + (-0.330 + 0.572i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0608 + 0.462i)T + (-18.3 + 4.91i)T^{2} \)
23 \( 1 + (1.23 + 0.330i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-3.73 + 9.01i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (4.28 - 7.42i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.88 - 2.21i)T + (9.57 + 35.7i)T^{2} \)
41 \( 1 + (-1.52 + 1.52i)T - 41iT^{2} \)
43 \( 1 + (7.84 - 3.25i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + (-0.625 + 0.361i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.75 - 0.888i)T + (51.1 + 13.7i)T^{2} \)
59 \( 1 + (1.04 - 7.90i)T + (-56.9 - 15.2i)T^{2} \)
61 \( 1 + (-0.352 - 2.67i)T + (-58.9 + 15.7i)T^{2} \)
67 \( 1 + (-1.83 + 1.41i)T + (17.3 - 64.7i)T^{2} \)
71 \( 1 + (5.29 + 5.29i)T + 71iT^{2} \)
73 \( 1 + (-1.53 - 5.71i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.78 - 6.55i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.37 - 1.81i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-4.09 + 15.2i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 - 11.4iT - 97T^{2} \)
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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

−11.99954691979017081595720346891, −11.81007805501110981570084303215, −10.33786547597964667161064020454, −9.723561022290373259335015556704, −8.930295477410012634146314034273, −7.44060192091078085501923941427, −6.59669731923400468210933995551, −4.54290376629273786601141563136, −3.57635171207817323215485549050, −2.42217152139139916204001087842, 0.54565587033689096795670142194, 3.48772139978790733783503954719, 4.87967642545654015577081564372, 5.86078446847513677682088178683, 7.27426602324131566143663856049, 8.048492650855127072345966862070, 8.709529224649076430562942320547, 9.965333302405984976040869766791, 10.86244102690370077937126894772, 12.52904102335431800967178991296

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