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Properties

Label 2-2e4-16.5-c7-0-11
Degree $2$
Conductor $16$
Sign $-0.860 + 0.509i$
Analytic cond. $4.99816$
Root an. cond. $2.23565$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.94 − 9.62i)2-s + (−12.2 + 12.2i)3-s + (−57.2 − 114. i)4-s + (−210. − 210. i)5-s + (45.0 + 190. i)6-s − 920. i·7-s + (−1.44e3 − 130. i)8-s + 1.88e3i·9-s + (−3.27e3 + 773. i)10-s + (−899. − 899. i)11-s + (2.10e3 + 702. i)12-s + (8.18e3 − 8.18e3i)13-s + (−8.85e3 − 5.47e3i)14-s + 5.16e3·15-s + (−9.83e3 + 1.31e4i)16-s + 2.46e4·17-s + ⋯
L(s)  = 1  + (0.525 − 0.850i)2-s + (−0.262 + 0.262i)3-s + (−0.447 − 0.894i)4-s + (−0.752 − 0.752i)5-s + (0.0851 + 0.360i)6-s − 1.01i·7-s + (−0.995 − 0.0899i)8-s + 0.862i·9-s + (−1.03 + 0.244i)10-s + (−0.203 − 0.203i)11-s + (0.351 + 0.117i)12-s + (1.03 − 1.03i)13-s + (−0.862 − 0.533i)14-s + 0.394·15-s + (−0.600 + 0.799i)16-s + 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $-0.860 + 0.509i$
Analytic conductor: \(4.99816\)
Root analytic conductor: \(2.23565\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 16,\ (\ :7/2),\ -0.860 + 0.509i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.349890 - 1.27820i\)
\(L(\frac12)\) \(\approx\) \(0.349890 - 1.27820i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.94 + 9.62i)T \)
good3 \( 1 + (12.2 - 12.2i)T - 2.18e3iT^{2} \)
5 \( 1 + (210. + 210. i)T + 7.81e4iT^{2} \)
7 \( 1 + 920. iT - 8.23e5T^{2} \)
11 \( 1 + (899. + 899. i)T + 1.94e7iT^{2} \)
13 \( 1 + (-8.18e3 + 8.18e3i)T - 6.27e7iT^{2} \)
17 \( 1 - 2.46e4T + 4.10e8T^{2} \)
19 \( 1 + (-5.11e3 + 5.11e3i)T - 8.93e8iT^{2} \)
23 \( 1 - 6.06e3iT - 3.40e9T^{2} \)
29 \( 1 + (-7.17e4 + 7.17e4i)T - 1.72e10iT^{2} \)
31 \( 1 + 2.85e5T + 2.75e10T^{2} \)
37 \( 1 + (-1.55e5 - 1.55e5i)T + 9.49e10iT^{2} \)
41 \( 1 - 4.74e5iT - 1.94e11T^{2} \)
43 \( 1 + (6.54e5 + 6.54e5i)T + 2.71e11iT^{2} \)
47 \( 1 - 1.21e6T + 5.06e11T^{2} \)
53 \( 1 + (4.39e5 + 4.39e5i)T + 1.17e12iT^{2} \)
59 \( 1 + (2.86e5 + 2.86e5i)T + 2.48e12iT^{2} \)
61 \( 1 + (-1.57e6 + 1.57e6i)T - 3.14e12iT^{2} \)
67 \( 1 + (-2.64e6 + 2.64e6i)T - 6.06e12iT^{2} \)
71 \( 1 - 2.51e6iT - 9.09e12T^{2} \)
73 \( 1 - 4.02e6iT - 1.10e13T^{2} \)
79 \( 1 - 4.77e6T + 1.92e13T^{2} \)
83 \( 1 + (4.69e6 - 4.69e6i)T - 2.71e13iT^{2} \)
89 \( 1 - 3.62e6iT - 4.42e13T^{2} \)
97 \( 1 + 2.99e6T + 8.07e13T^{2} \)
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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

−16.79373762423690709093315005978, −15.64013642933560528626456323518, −13.85659146106738517720324343309, −12.77429082226725999082757378707, −11.26377557997754253915784967957, −10.21696261214832713362622408183, −8.099943327060184334064502048879, −5.29493763849904403604576755531, −3.76639933689122197771596138914, −0.77288600578589053472681184095, 3.55187823194222877065413316151, 5.87836074895028282495153030563, 7.25132153359175380427176608689, 8.949771236207362555102151388945, 11.54560823269770326013075898417, 12.51804315774018905329770350183, 14.36560181327238702702375783433, 15.29756929217223508655275882006, 16.41861817387300753135937306701, 18.17051691264712302191035241711

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