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Properties

Label 2-224-224.115-c1-0-12
Degree $2$
Conductor $224$
Sign $0.686 + 0.727i$
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  + (−1.17 + 0.780i)2-s + (−0.638 − 0.832i)3-s + (0.782 − 1.84i)4-s + (0.171 − 0.222i)5-s + (1.40 + 0.483i)6-s + (−0.686 + 2.55i)7-s + (0.512 + 2.78i)8-s + (0.491 − 1.83i)9-s + (−0.0278 + 0.396i)10-s + (0.596 − 4.53i)11-s + (−2.03 + 0.523i)12-s + (0.625 − 1.50i)13-s + (−1.18 − 3.54i)14-s − 0.294·15-s + (−2.77 − 2.88i)16-s + (3.61 − 6.26i)17-s + ⋯
L(s)  = 1  + (−0.834 + 0.551i)2-s + (−0.368 − 0.480i)3-s + (0.391 − 0.920i)4-s + (0.0765 − 0.0997i)5-s + (0.572 + 0.197i)6-s + (−0.259 + 0.965i)7-s + (0.181 + 0.983i)8-s + (0.163 − 0.611i)9-s + (−0.00881 + 0.125i)10-s + (0.179 − 1.36i)11-s + (−0.586 + 0.151i)12-s + (0.173 − 0.418i)13-s + (−0.316 − 0.948i)14-s − 0.0761·15-s + (−0.693 − 0.720i)16-s + (0.877 − 1.51i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.686 + 0.727i$
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: \((2,\ 224,\ (\ :1/2),\ 0.686 + 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.653712 - 0.281838i\)
\(L(\frac12)\) \(\approx\) \(0.653712 - 0.281838i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 - 0.780i)T \)
7 \( 1 + (0.686 - 2.55i)T \)
good3 \( 1 + (0.638 + 0.832i)T + (-0.776 + 2.89i)T^{2} \)
5 \( 1 + (-0.171 + 0.222i)T + (-1.29 - 4.82i)T^{2} \)
11 \( 1 + (-0.596 + 4.53i)T + (-10.6 - 2.84i)T^{2} \)
13 \( 1 + (-0.625 + 1.50i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + (-3.61 + 6.26i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0661 + 0.502i)T + (-18.3 + 4.91i)T^{2} \)
23 \( 1 + (-1.64 - 0.441i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-3.42 + 8.27i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (0.330 - 0.571i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.974 + 0.748i)T + (9.57 + 35.7i)T^{2} \)
41 \( 1 + (2.88 - 2.88i)T - 41iT^{2} \)
43 \( 1 + (-3.20 + 1.32i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + (10.8 - 6.26i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.08 - 1.19i)T + (51.1 + 13.7i)T^{2} \)
59 \( 1 + (-0.651 + 4.94i)T + (-56.9 - 15.2i)T^{2} \)
61 \( 1 + (-1.58 - 12.0i)T + (-58.9 + 15.7i)T^{2} \)
67 \( 1 + (0.783 - 0.600i)T + (17.3 - 64.7i)T^{2} \)
71 \( 1 + (-5.15 - 5.15i)T + 71iT^{2} \)
73 \( 1 + (-2.30 - 8.59i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.927 + 1.60i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.7 + 4.44i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (1.99 - 7.44i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 - 1.92iT - 97T^{2} \)
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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

−11.78438344533982665268987315762, −11.36093555219805617679060182281, −9.881715074809580272069666135095, −9.133911244755073256331454609438, −8.221181684565879492881289973068, −7.05284475911005136481379660519, −6.04299746917574165260324472632, −5.37893829895311600337056728990, −2.97184300344301491161438540876, −0.877049255024749245544443786736, 1.73410603352661229133127222095, 3.65710123390106362859853403246, 4.70269889640753613602850169578, 6.57807996019974494609795711577, 7.47054561567691189821694143630, 8.545173154864975126657902137149, 9.977819239614758745319798373309, 10.23137838515897955754196793077, 11.06774524060309623339323711212, 12.29066476696938253043683751709

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