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Properties

Label 2-208-13.10-c3-0-5
Degree $2$
Conductor $208$
Sign $-0.538 - 0.842i$
Analytic cond. $12.2723$
Root an. cond. $3.50319$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.96 + 8.59i)3-s − 13.0i·5-s + (6.90 + 3.98i)7-s + (−35.7 + 61.8i)9-s + (−46.0 + 26.5i)11-s + (2.73 + 46.7i)13-s + (112. − 64.8i)15-s + (−14.9 + 25.8i)17-s + (126. + 72.9i)19-s + 79.1i·21-s + (−0.458 − 0.794i)23-s − 46.0·25-s − 440.·27-s + (−31.5 − 54.5i)29-s + 136. i·31-s + ⋯
L(s)  = 1  + (0.954 + 1.65i)3-s − 1.16i·5-s + (0.372 + 0.215i)7-s + (−1.32 + 2.29i)9-s + (−1.26 + 0.728i)11-s + (0.0583 + 0.998i)13-s + (1.93 − 1.11i)15-s + (−0.212 + 0.368i)17-s + (1.52 + 0.880i)19-s + 0.822i·21-s + (−0.00415 − 0.00720i)23-s − 0.368·25-s − 3.14·27-s + (−0.201 − 0.349i)29-s + 0.791i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.538 - 0.842i$
Analytic conductor: \(12.2723\)
Root analytic conductor: \(3.50319\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :3/2),\ -0.538 - 0.842i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.00595 + 1.83775i\)
\(L(\frac12)\) \(\approx\) \(1.00595 + 1.83775i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.73 - 46.7i)T \)
good3 \( 1 + (-4.96 - 8.59i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 13.0iT - 125T^{2} \)
7 \( 1 + (-6.90 - 3.98i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (46.0 - 26.5i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (14.9 - 25.8i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-126. - 72.9i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (0.458 + 0.794i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (31.5 + 54.5i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 136. iT - 2.97e4T^{2} \)
37 \( 1 + (-165. + 95.4i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (18.9 - 10.9i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-167. + 290. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 52.8iT - 1.03e5T^{2} \)
53 \( 1 + 275.T + 1.48e5T^{2} \)
59 \( 1 + (-353. - 203. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-401. + 696. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-205. + 118. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-111. - 64.1i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 287. iT - 3.89e5T^{2} \)
79 \( 1 - 1.09e3T + 4.93e5T^{2} \)
83 \( 1 - 16.7iT - 5.71e5T^{2} \)
89 \( 1 + (-77.2 + 44.5i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-1.00e3 - 581. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.25890791276445861822488060156, −11.06848434263661188080831140562, −10.00511632220919321364016072587, −9.369038984329294247652878130624, −8.505362729650884035988603120263, −7.73365869985761782542788115428, −5.33358748813142378068231877256, −4.79962388002959982869181553796, −3.70331036541054266034161642155, −2.13185385785030658973299813740, 0.798439851824376193744844492744, 2.63954848543112628729575798578, 3.10344782581002007844855975943, 5.60091406089446983435088738078, 6.78396879709475646847879025044, 7.64736174453014747124404040970, 8.101046966408355244399801012512, 9.476169260503484212213014875042, 10.86781867201119983154730593058, 11.60799770188265753324269684154

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