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Properties

Label 2-208-13.11-c2-0-7
Degree $2$
Conductor $208$
Sign $0.715 + 0.698i$
Analytic cond. $5.66758$
Root an. cond. $2.38066$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0172 + 0.0298i)3-s + (2.08 + 2.08i)5-s + (−2.54 − 9.49i)7-s + (4.49 − 7.79i)9-s + (−3.98 − 1.06i)11-s + (12.4 + 3.69i)13-s + (−0.0262 + 0.0980i)15-s + (5.17 + 2.98i)17-s + (27.3 − 7.32i)19-s + (0.239 − 0.239i)21-s + (−0.0356 + 0.0205i)23-s − 16.3i·25-s + 0.619·27-s + (−19.8 − 34.3i)29-s + (9.51 + 9.51i)31-s + ⋯
L(s)  = 1  + (0.00573 + 0.00993i)3-s + (0.417 + 0.417i)5-s + (−0.363 − 1.35i)7-s + (0.499 − 0.865i)9-s + (−0.362 − 0.0971i)11-s + (0.958 + 0.283i)13-s + (−0.00175 + 0.00653i)15-s + (0.304 + 0.175i)17-s + (1.43 − 0.385i)19-s + (0.0113 − 0.0113i)21-s + (−0.00155 + 0.000895i)23-s − 0.652i·25-s + 0.0229·27-s + (−0.684 − 1.18i)29-s + (0.306 + 0.306i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.715 + 0.698i$
Analytic conductor: \(5.66758\)
Root analytic conductor: \(2.38066\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :1),\ 0.715 + 0.698i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.50996 - 0.614400i\)
\(L(\frac12)\) \(\approx\) \(1.50996 - 0.614400i\)
\(L(2)\) 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 \( 1 + (-12.4 - 3.69i)T \)
good3 \( 1 + (-0.0172 - 0.0298i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-2.08 - 2.08i)T + 25iT^{2} \)
7 \( 1 + (2.54 + 9.49i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (3.98 + 1.06i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (-5.17 - 2.98i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-27.3 + 7.32i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (0.0356 - 0.0205i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (19.8 + 34.3i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-9.51 - 9.51i)T + 961iT^{2} \)
37 \( 1 + (21.9 + 5.88i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (7.38 - 27.5i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-37.5 - 21.6i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (56.6 - 56.6i)T - 2.20e3iT^{2} \)
53 \( 1 - 30.6T + 2.80e3T^{2} \)
59 \( 1 + (-7.88 - 29.4i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (43.6 - 75.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-15.7 + 58.6i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (95.2 - 25.5i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (-35.7 + 35.7i)T - 5.32e3iT^{2} \)
79 \( 1 + 79.3T + 6.24e3T^{2} \)
83 \( 1 + (-52.3 - 52.3i)T + 6.88e3iT^{2} \)
89 \( 1 + (-144. - 38.6i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (175. - 47.1i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−12.00793091045078192343045791384, −10.92459237655112983658781621584, −10.07731634874350659218780002120, −9.346169970525516206364495586216, −7.83013554839511723441422232930, −6.87700049034615334690285473863, −5.98904234248289806495809568845, −4.27174909096556456664964238259, −3.21633878475543988769066244120, −1.04745025418349689970263815212, 1.74802979697711804425460510122, 3.27652629363010664224376944755, 5.19013583740708812870067661137, 5.70949261659129792151159877671, 7.25697129149010524888860647933, 8.418221483891605306703791344088, 9.294044651216880412256475316726, 10.22539846549132046783565710673, 11.37550720088176658012208542204, 12.40453795789402289052368381905

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