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Properties

Label 2-208-16.13-c1-0-11
Degree $2$
Conductor $208$
Sign $-0.213 + 0.976i$
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 + 0.0627i)2-s + (−1.56 − 1.56i)3-s + (1.99 − 0.177i)4-s + (3.08 − 3.08i)5-s + (2.30 + 2.10i)6-s + 2.75i·7-s + (−2.80 + 0.375i)8-s + 1.87i·9-s + (−4.16 + 4.55i)10-s + (2.49 − 2.49i)11-s + (−3.38 − 2.83i)12-s + (−0.707 − 0.707i)13-s + (−0.173 − 3.89i)14-s − 9.63·15-s + (3.93 − 0.706i)16-s − 0.00475·17-s + ⋯
L(s)  = 1  + (−0.999 + 0.0443i)2-s + (−0.900 − 0.900i)3-s + (0.996 − 0.0886i)4-s + (1.38 − 1.38i)5-s + (0.940 + 0.860i)6-s + 1.04i·7-s + (−0.991 + 0.132i)8-s + 0.623i·9-s + (−1.31 + 1.44i)10-s + (0.753 − 0.753i)11-s + (−0.977 − 0.817i)12-s + (−0.196 − 0.196i)13-s + (−0.0462 − 1.04i)14-s − 2.48·15-s + (0.984 − 0.176i)16-s − 0.00115·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.213 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $-0.213 + 0.976i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :1/2),\ -0.213 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.448287 - 0.556850i\)
\(L(\frac12)\) \(\approx\) \(0.448287 - 0.556850i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0627i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (1.56 + 1.56i)T + 3iT^{2} \)
5 \( 1 + (-3.08 + 3.08i)T - 5iT^{2} \)
7 \( 1 - 2.75iT - 7T^{2} \)
11 \( 1 + (-2.49 + 2.49i)T - 11iT^{2} \)
17 \( 1 + 0.00475T + 17T^{2} \)
19 \( 1 + (3.46 + 3.46i)T + 19iT^{2} \)
23 \( 1 - 0.520iT - 23T^{2} \)
29 \( 1 + (-1.10 - 1.10i)T + 29iT^{2} \)
31 \( 1 + 1.86T + 31T^{2} \)
37 \( 1 + (5.05 - 5.05i)T - 37iT^{2} \)
41 \( 1 - 4.45iT - 41T^{2} \)
43 \( 1 + (-0.191 + 0.191i)T - 43iT^{2} \)
47 \( 1 - 7.37T + 47T^{2} \)
53 \( 1 + (-8.72 + 8.72i)T - 53iT^{2} \)
59 \( 1 + (2.26 - 2.26i)T - 59iT^{2} \)
61 \( 1 + (-3.55 - 3.55i)T + 61iT^{2} \)
67 \( 1 + (-7.35 - 7.35i)T + 67iT^{2} \)
71 \( 1 + 9.99iT - 71T^{2} \)
73 \( 1 - 12.9iT - 73T^{2} \)
79 \( 1 - 9.41T + 79T^{2} \)
83 \( 1 + (4.61 + 4.61i)T + 83iT^{2} \)
89 \( 1 - 9.34iT - 89T^{2} \)
97 \( 1 + 16.0T + 97T^{2} \)
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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

−12.14641493256438697760515500526, −11.27162629076562437735654458666, −9.950908481058224294183740200058, −8.945664596898890427928466694424, −8.524455314911320763545179223916, −6.77322794625334599583295824970, −5.97792335974064292459839321066, −5.31585177897178039735293112152, −2.19471583206622659650394409167, −0.980644349785697082549488707820, 2.04770693833414343441395241614, 3.88503029239102026578736393743, 5.68384125054779437074280279175, 6.60418963970812201806942715049, 7.34339798593629639257951246293, 9.240428458922507218116214642011, 10.04657474316452925834468460308, 10.52146585756730599857217420147, 11.04680563252286660461026478580, 12.28241761991987365415743922570

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