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Properties

Label 2-208-13.12-c1-0-2
Degree $2$
Conductor $208$
Sign $0.987 - 0.155i$
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  + 2.56·3-s + 2.56i·5-s − 4.56i·7-s + 3.56·9-s + 3.12i·11-s + (−3.56 + 0.561i)13-s + 6.56i·15-s + 0.561·17-s − 2i·19-s − 11.6i·21-s − 5.12·23-s − 1.56·25-s + 1.43·27-s + 3.12·29-s − 3.12i·31-s + ⋯
L(s)  = 1  + 1.47·3-s + 1.14i·5-s − 1.72i·7-s + 1.18·9-s + 0.941i·11-s + (−0.987 + 0.155i)13-s + 1.69i·15-s + 0.136·17-s − 0.458i·19-s − 2.54i·21-s − 1.06·23-s − 0.312·25-s + 0.276·27-s + 0.579·29-s − 0.560i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(208\)    =    \(2^{4} \cdot 13\)
Sign: $0.987 - 0.155i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{208} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 208,\ (\ :1/2),\ 0.987 - 0.155i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76355 + 0.138176i\)
\(L(\frac12)\) \(\approx\) \(1.76355 + 0.138176i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (3.56 - 0.561i)T \)
good3 \( 1 - 2.56T + 3T^{2} \)
5 \( 1 - 2.56iT - 5T^{2} \)
7 \( 1 + 4.56iT - 7T^{2} \)
11 \( 1 - 3.12iT - 11T^{2} \)
17 \( 1 - 0.561T + 17T^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + 5.12T + 23T^{2} \)
29 \( 1 - 3.12T + 29T^{2} \)
31 \( 1 + 3.12iT - 31T^{2} \)
37 \( 1 - 5.43iT - 37T^{2} \)
41 \( 1 + 8iT - 41T^{2} \)
43 \( 1 + 5.43T + 43T^{2} \)
47 \( 1 - 3.43iT - 47T^{2} \)
53 \( 1 + 4.24T + 53T^{2} \)
59 \( 1 + 10iT - 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 - 0.876iT - 67T^{2} \)
71 \( 1 - 5.68iT - 71T^{2} \)
73 \( 1 - 15.3iT - 73T^{2} \)
79 \( 1 - 2.87T + 79T^{2} \)
83 \( 1 - 8.24iT - 83T^{2} \)
89 \( 1 - 5.12iT - 89T^{2} \)
97 \( 1 + 10.2iT - 97T^{2} \)
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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

−12.67809661114802735962168137405, −11.25609523321771988466309558011, −10.01332572039276304064827453166, −9.865949096995231277336813937648, −8.207271747403520063644408107762, −7.31309324514304342352421951481, −6.84873395150265571987361621358, −4.47554647673368164616675853446, −3.46540152404583342168904368140, −2.24780562246778179626587056746, 2.08938866182824901652739459135, 3.24493170084955650768751205588, 4.88024002068977209787611849174, 5.96763607290205902632952773588, 7.85490991353646313061301191892, 8.561093818661276367832808065335, 9.053255061673221824857940591604, 9.969519716362535778672141063721, 11.79170467711463910852945495137, 12.43824145896096967470392876128

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