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Properties

Label 2-208-16.13-c1-0-1
Degree $2$
Conductor $208$
Sign $0.684 - 0.729i$
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 − 0.127i)2-s + (−2.17 − 2.17i)3-s + (1.96 + 0.358i)4-s + (−2.25 + 2.25i)5-s + (2.78 + 3.33i)6-s − 2.10i·7-s + (−2.72 − 0.755i)8-s + 6.44i·9-s + (3.45 − 2.88i)10-s + (−0.970 + 0.970i)11-s + (−3.49 − 5.05i)12-s + (0.707 + 0.707i)13-s + (−0.268 + 2.96i)14-s + 9.78·15-s + (3.74 + 1.41i)16-s + 5.32·17-s + ⋯
L(s)  = 1  + (−0.995 − 0.0900i)2-s + (−1.25 − 1.25i)3-s + (0.983 + 0.179i)4-s + (−1.00 + 1.00i)5-s + (1.13 + 1.36i)6-s − 0.796i·7-s + (−0.963 − 0.267i)8-s + 2.14i·9-s + (1.09 − 0.912i)10-s + (−0.292 + 0.292i)11-s + (−1.00 − 1.45i)12-s + (0.196 + 0.196i)13-s + (−0.0717 + 0.793i)14-s + 2.52·15-s + (0.935 + 0.352i)16-s + 1.29·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 - 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.684 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.684 - 0.729i$
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: \((2,\ 208,\ (\ :1/2),\ 0.684 - 0.729i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.278435 + 0.120605i\)
\(L(\frac12)\) \(\approx\) \(0.278435 + 0.120605i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.127i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (2.17 + 2.17i)T + 3iT^{2} \)
5 \( 1 + (2.25 - 2.25i)T - 5iT^{2} \)
7 \( 1 + 2.10iT - 7T^{2} \)
11 \( 1 + (0.970 - 0.970i)T - 11iT^{2} \)
17 \( 1 - 5.32T + 17T^{2} \)
19 \( 1 + (-4.19 - 4.19i)T + 19iT^{2} \)
23 \( 1 - 4.81iT - 23T^{2} \)
29 \( 1 + (3.09 + 3.09i)T + 29iT^{2} \)
31 \( 1 + 4.15T + 31T^{2} \)
37 \( 1 + (6.92 - 6.92i)T - 37iT^{2} \)
41 \( 1 - 9.94iT - 41T^{2} \)
43 \( 1 + (1.27 - 1.27i)T - 43iT^{2} \)
47 \( 1 - 0.528T + 47T^{2} \)
53 \( 1 + (0.242 - 0.242i)T - 53iT^{2} \)
59 \( 1 + (1.95 - 1.95i)T - 59iT^{2} \)
61 \( 1 + (-5.73 - 5.73i)T + 61iT^{2} \)
67 \( 1 + (6.99 + 6.99i)T + 67iT^{2} \)
71 \( 1 - 1.27iT - 71T^{2} \)
73 \( 1 - 3.73iT - 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + (-10.0 - 10.0i)T + 83iT^{2} \)
89 \( 1 + 9.75iT - 89T^{2} \)
97 \( 1 - 4.87T + 97T^{2} \)
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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

−11.88926607163057393053227178454, −11.68169099465318191521165401007, −10.71750968362560231811292308248, −9.970747004884656513775571801413, −7.81713624546164303166133619383, −7.57610584153980122348120795640, −6.77924916766656336545533937835, −5.65762926676295965540577287586, −3.40445004098831697046717063008, −1.34877392478787223453230785520, 0.46055977236967054098552435818, 3.54369265952681175167217838804, 5.13191672077378193226538501300, 5.69823011910183384113178882124, 7.32066390511260113341062173229, 8.654132055523451185245017026111, 9.238705788730296682685129876754, 10.38542921351062067245813724598, 11.15899813994800263746104452901, 12.02259907274893317554287860681

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