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Properties

Label 2-208-13.12-c9-0-13
Degree $2$
Conductor $208$
Sign $0.0618 - 0.998i$
Analytic cond. $107.127$
Root an. cond. $10.3502$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 263.·3-s − 382. i·5-s − 2.90e3i·7-s + 4.99e4·9-s + 4.66e4i·11-s + (−6.36e3 + 1.02e5i)13-s + 1.01e5i·15-s + 9.66e4·17-s + 2.12e5i·19-s + 7.65e5i·21-s − 1.18e6·23-s + 1.80e6·25-s − 7.98e6·27-s + 3.10e6·29-s − 2.63e6i·31-s + ⋯
L(s)  = 1  − 1.88·3-s − 0.273i·5-s − 0.456i·7-s + 2.53·9-s + 0.960i·11-s + (−0.0618 + 0.998i)13-s + 0.515i·15-s + 0.280·17-s + 0.374i·19-s + 0.858i·21-s − 0.882·23-s + 0.924·25-s − 2.89·27-s + 0.814·29-s − 0.511i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0618 - 0.998i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.0618 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.0618 - 0.998i$
Analytic conductor: \(107.127\)
Root analytic conductor: \(10.3502\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :9/2),\ 0.0618 - 0.998i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.7125049741\)
\(L(\frac12)\) \(\approx\) \(0.7125049741\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (6.36e3 - 1.02e5i)T \)
good3 \( 1 + 263.T + 1.96e4T^{2} \)
5 \( 1 + 382. iT - 1.95e6T^{2} \)
7 \( 1 + 2.90e3iT - 4.03e7T^{2} \)
11 \( 1 - 4.66e4iT - 2.35e9T^{2} \)
17 \( 1 - 9.66e4T + 1.18e11T^{2} \)
19 \( 1 - 2.12e5iT - 3.22e11T^{2} \)
23 \( 1 + 1.18e6T + 1.80e12T^{2} \)
29 \( 1 - 3.10e6T + 1.45e13T^{2} \)
31 \( 1 + 2.63e6iT - 2.64e13T^{2} \)
37 \( 1 + 1.62e7iT - 1.29e14T^{2} \)
41 \( 1 - 1.26e7iT - 3.27e14T^{2} \)
43 \( 1 - 1.44e7T + 5.02e14T^{2} \)
47 \( 1 + 2.14e7iT - 1.11e15T^{2} \)
53 \( 1 + 3.61e7T + 3.29e15T^{2} \)
59 \( 1 - 2.51e7iT - 8.66e15T^{2} \)
61 \( 1 + 1.25e8T + 1.16e16T^{2} \)
67 \( 1 - 1.39e8iT - 2.72e16T^{2} \)
71 \( 1 + 3.64e8iT - 4.58e16T^{2} \)
73 \( 1 + 4.77e8iT - 5.88e16T^{2} \)
79 \( 1 + 6.04e8T + 1.19e17T^{2} \)
83 \( 1 - 4.13e8iT - 1.86e17T^{2} \)
89 \( 1 - 7.43e8iT - 3.50e17T^{2} \)
97 \( 1 - 1.20e9iT - 7.60e17T^{2} \)
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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

−10.97820662765688131806422016731, −10.25836514067763017596177717271, −9.354620417181562825063244547830, −7.60395790954114319498299304654, −6.76498390384865371857736343484, −5.88048723087019528094210932198, −4.76001274269278174920396937632, −4.13336155220768278602789231372, −1.83644181352623082068258032336, −0.798072541583474882618554829238, 0.30301050708747177385077436051, 1.20677627037979392791005958366, 3.00678972671735726571985283496, 4.54971767563451757021992127744, 5.55874824175754713591106374210, 6.14718797771155831602668333819, 7.17315761067870460939367951327, 8.468543164935703367121251360786, 9.981175034896312732365869335428, 10.66604409568314448242109393279

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