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Properties

Label 2-208-16.13-c1-0-13
Degree $2$
Conductor $208$
Sign $0.430 - 0.902i$
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  + (0.986 + 1.01i)2-s + (0.718 + 0.718i)3-s + (−0.0523 + 1.99i)4-s + (2.02 − 2.02i)5-s + (−0.0188 + 1.43i)6-s + 0.407i·7-s + (−2.07 + 1.91i)8-s − 1.96i·9-s + (4.04 + 0.0529i)10-s + (−1.76 + 1.76i)11-s + (−1.47 + 1.39i)12-s + (−0.707 − 0.707i)13-s + (−0.413 + 0.402i)14-s + 2.90·15-s + (−3.99 − 0.209i)16-s − 4.09·17-s + ⋯
L(s)  = 1  + (0.697 + 0.716i)2-s + (0.414 + 0.414i)3-s + (−0.0261 + 0.999i)4-s + (0.903 − 0.903i)5-s + (−0.00767 + 0.586i)6-s + 0.154i·7-s + (−0.734 + 0.678i)8-s − 0.656i·9-s + (1.27 + 0.0167i)10-s + (−0.530 + 0.530i)11-s + (−0.425 + 0.403i)12-s + (−0.196 − 0.196i)13-s + (−0.110 + 0.107i)14-s + 0.749·15-s + (−0.998 − 0.0523i)16-s − 0.991·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.430 - 0.902i$
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: \((2,\ 208,\ (\ :1/2),\ 0.430 - 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67803 + 1.05873i\)
\(L(\frac12)\) \(\approx\) \(1.67803 + 1.05873i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.986 - 1.01i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-0.718 - 0.718i)T + 3iT^{2} \)
5 \( 1 + (-2.02 + 2.02i)T - 5iT^{2} \)
7 \( 1 - 0.407iT - 7T^{2} \)
11 \( 1 + (1.76 - 1.76i)T - 11iT^{2} \)
17 \( 1 + 4.09T + 17T^{2} \)
19 \( 1 + (2.13 + 2.13i)T + 19iT^{2} \)
23 \( 1 - 2.42iT - 23T^{2} \)
29 \( 1 + (-2.51 - 2.51i)T + 29iT^{2} \)
31 \( 1 - 0.736T + 31T^{2} \)
37 \( 1 + (-1.13 + 1.13i)T - 37iT^{2} \)
41 \( 1 + 10.6iT - 41T^{2} \)
43 \( 1 + (0.341 - 0.341i)T - 43iT^{2} \)
47 \( 1 - 11.4T + 47T^{2} \)
53 \( 1 + (6.62 - 6.62i)T - 53iT^{2} \)
59 \( 1 + (9.01 - 9.01i)T - 59iT^{2} \)
61 \( 1 + (2.36 + 2.36i)T + 61iT^{2} \)
67 \( 1 + (-0.526 - 0.526i)T + 67iT^{2} \)
71 \( 1 - 7.15iT - 71T^{2} \)
73 \( 1 - 5.01iT - 73T^{2} \)
79 \( 1 - 9.29T + 79T^{2} \)
83 \( 1 + (10.8 + 10.8i)T + 83iT^{2} \)
89 \( 1 + 11.8iT - 89T^{2} \)
97 \( 1 - 18.7T + 97T^{2} \)
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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

−12.78023716284405065548694840940, −12.05153746564778327035020707963, −10.49846706730961714610273179013, −9.132090591810243924606254449126, −8.857666466447713175393401848882, −7.39378400356823674547988306286, −6.17290796140356723979294682004, −5.14394552355198416326221890581, −4.17234299138492228428824001302, −2.52841996628236417998525214827, 2.06370230788074642376650273351, 2.89689492437296955638655692324, 4.59471797378318886663878896962, 5.95414752017975700877798072951, 6.80534730328314772793401283975, 8.246426318053333839244057813048, 9.581160701277001433824528982387, 10.58077426189943406053643653238, 11.02624608882342590076061499114, 12.41327857081283878576774095658

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