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Properties

Label 2-208-208.29-c1-0-1
Degree $2$
Conductor $208$
Sign $-0.763 + 0.645i$
Analytic cond. $1.66088$
Root an. cond. $1.28875$
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  + (−1.41 + 0.103i)2-s + (−0.623 + 2.32i)3-s + (1.97 − 0.291i)4-s + (−2.33 + 2.33i)5-s + (0.639 − 3.34i)6-s + (−3.16 − 1.82i)7-s + (−2.76 + 0.615i)8-s + (−2.43 − 1.40i)9-s + (3.05 − 3.53i)10-s + (3.60 + 0.965i)11-s + (−0.555 + 4.78i)12-s + (0.897 − 3.49i)13-s + (4.64 + 2.24i)14-s + (−3.98 − 6.90i)15-s + (3.83 − 1.15i)16-s + (1.39 − 2.41i)17-s + ⋯
L(s)  = 1  + (−0.997 + 0.0730i)2-s + (−0.360 + 1.34i)3-s + (0.989 − 0.145i)4-s + (−1.04 + 1.04i)5-s + (0.260 − 1.36i)6-s + (−1.19 − 0.690i)7-s + (−0.976 + 0.217i)8-s + (−0.810 − 0.467i)9-s + (0.966 − 1.11i)10-s + (1.08 + 0.291i)11-s + (−0.160 + 1.38i)12-s + (0.248 − 0.968i)13-s + (1.24 + 0.601i)14-s + (−1.02 − 1.78i)15-s + (0.957 − 0.288i)16-s + (0.337 − 0.585i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.763 + 0.645i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.763 + 0.645i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :1/2),\ -0.763 + 0.645i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0661943 - 0.180869i\)
\(L(\frac12)\) \(\approx\) \(0.0661943 - 0.180869i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.103i)T \)
13 \( 1 + (-0.897 + 3.49i)T \)
good3 \( 1 + (0.623 - 2.32i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (2.33 - 2.33i)T - 5iT^{2} \)
7 \( 1 + (3.16 + 1.82i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.60 - 0.965i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.39 + 2.41i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.63 - 1.77i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (5.70 - 3.29i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.314 - 1.17i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + 4.09T + 31T^{2} \)
37 \( 1 + (-3.69 - 0.988i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.123 - 0.0712i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.772 - 2.88i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + 7.34T + 47T^{2} \)
53 \( 1 + (6.32 - 6.32i)T - 53iT^{2} \)
59 \( 1 + (0.387 + 1.44i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.41 - 0.646i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (0.625 - 2.33i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-7.89 - 4.56i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 10.8iT - 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + (-0.331 - 0.331i)T + 83iT^{2} \)
89 \( 1 + (6.75 - 3.89i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.70 + 9.87i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.60191346874674029104463820732, −11.46781579856001767205757252811, −10.78962248742221449990190034490, −10.07046041168523502856988955929, −9.449489975535783296543062616619, −8.010701330719739920244574215839, −6.96488610554348474597890860396, −6.05755490522324465963876027432, −3.98916652939094681512662415204, −3.28947126669038453373718656285, 0.22754630844242398635705951191, 1.86773967186274459493529455431, 3.88782610756310325039351361265, 6.23616117516151612374934309262, 6.57371034364015069408182173115, 7.931167373483529146741037109578, 8.687183980055173431972424714572, 9.463263107299208887663888954345, 11.10329671841825950613954770484, 12.11372393687752654212693875234

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