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Properties

Label 2-208-16.13-c1-0-15
Degree $2$
Conductor $208$
Sign $0.810 - 0.585i$
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  + (1.40 + 0.194i)2-s + (0.902 + 0.902i)3-s + (1.92 + 0.545i)4-s + (−1.29 + 1.29i)5-s + (1.08 + 1.44i)6-s − 3.20i·7-s + (2.58 + 1.13i)8-s − 1.36i·9-s + (−2.06 + 1.56i)10-s + (−2.72 + 2.72i)11-s + (1.24 + 2.22i)12-s + (0.707 + 0.707i)13-s + (0.623 − 4.49i)14-s − 2.33·15-s + (3.40 + 2.09i)16-s − 6.03·17-s + ⋯
L(s)  = 1  + (0.990 + 0.137i)2-s + (0.521 + 0.521i)3-s + (0.962 + 0.272i)4-s + (−0.579 + 0.579i)5-s + (0.444 + 0.587i)6-s − 1.21i·7-s + (0.915 + 0.402i)8-s − 0.456i·9-s + (−0.653 + 0.493i)10-s + (−0.820 + 0.820i)11-s + (0.359 + 0.643i)12-s + (0.196 + 0.196i)13-s + (0.166 − 1.20i)14-s − 0.603·15-s + (0.851 + 0.524i)16-s − 1.46·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 - 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.810 - 0.585i$
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: \((2,\ 208,\ (\ :1/2),\ 0.810 - 0.585i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08672 + 0.675447i\)
\(L(\frac12)\) \(\approx\) \(2.08672 + 0.675447i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.194i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-0.902 - 0.902i)T + 3iT^{2} \)
5 \( 1 + (1.29 - 1.29i)T - 5iT^{2} \)
7 \( 1 + 3.20iT - 7T^{2} \)
11 \( 1 + (2.72 - 2.72i)T - 11iT^{2} \)
17 \( 1 + 6.03T + 17T^{2} \)
19 \( 1 + (-1.89 - 1.89i)T + 19iT^{2} \)
23 \( 1 + 8.67iT - 23T^{2} \)
29 \( 1 + (6.82 + 6.82i)T + 29iT^{2} \)
31 \( 1 - 5.23T + 31T^{2} \)
37 \( 1 + (0.208 - 0.208i)T - 37iT^{2} \)
41 \( 1 - 9.29iT - 41T^{2} \)
43 \( 1 + (-0.0416 + 0.0416i)T - 43iT^{2} \)
47 \( 1 - 2.16T + 47T^{2} \)
53 \( 1 + (6.38 - 6.38i)T - 53iT^{2} \)
59 \( 1 + (-3.38 + 3.38i)T - 59iT^{2} \)
61 \( 1 + (-1.72 - 1.72i)T + 61iT^{2} \)
67 \( 1 + (1.56 + 1.56i)T + 67iT^{2} \)
71 \( 1 - 9.73iT - 71T^{2} \)
73 \( 1 - 6.08iT - 73T^{2} \)
79 \( 1 + 5.31T + 79T^{2} \)
83 \( 1 + (-9.71 - 9.71i)T + 83iT^{2} \)
89 \( 1 + 5.94iT - 89T^{2} \)
97 \( 1 + 9.79T + 97T^{2} \)
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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

−12.69990080699126453427244904337, −11.48574589120231173148576927621, −10.69505989587216724892635400375, −9.824335794477544330197929290152, −8.198907685756644483288745443232, −7.24337384754522323232994056936, −6.41304007111583595805115688205, −4.53184211374883340427176876811, −3.97136947479446954180658082018, −2.66804591653119415943655516621, 2.10405322380719833361892214314, 3.28793782580868011973713958353, 4.92974648576160206240482689629, 5.73667254531362582878172995023, 7.21742269708358580298986548571, 8.200661364931145808970791710304, 9.064255485614458178020559967773, 10.80242745251010153450799025974, 11.54465086433889382338907294691, 12.49328348868309450163085266054

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