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Properties

Label 2-1308-327.134-c0-0-0
Degree $2$
Conductor $1308$
Sign $0.985 + 0.172i$
Analytic cond. $0.652777$
Root an. cond. $0.807946$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.835 − 0.549i)3-s + (−0.0694 − 0.0932i)7-s + (0.396 + 0.918i)9-s + (0.914 + 1.22i)13-s + (−0.0201 + 0.114i)19-s + (0.00676 + 0.116i)21-s + (0.973 − 0.230i)25-s + (0.173 − 0.984i)27-s + (0.914 − 1.22i)31-s + (−0.786 − 0.0919i)37-s + (−0.0890 − 1.52i)39-s + (1.49 + 1.25i)43-s + (0.282 − 0.945i)49-s + (0.0798 − 0.0845i)57-s + (0.770 + 0.182i)61-s + ⋯
L(s)  = 1  + (−0.835 − 0.549i)3-s + (−0.0694 − 0.0932i)7-s + (0.396 + 0.918i)9-s + (0.914 + 1.22i)13-s + (−0.0201 + 0.114i)19-s + (0.00676 + 0.116i)21-s + (0.973 − 0.230i)25-s + (0.173 − 0.984i)27-s + (0.914 − 1.22i)31-s + (−0.786 − 0.0919i)37-s + (−0.0890 − 1.52i)39-s + (1.49 + 1.25i)43-s + (0.282 − 0.945i)49-s + (0.0798 − 0.0845i)57-s + (0.770 + 0.182i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1308\)    =    \(2^{2} \cdot 3 \cdot 109\)
Sign: $0.985 + 0.172i$
Analytic conductor: \(0.652777\)
Root analytic conductor: \(0.807946\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1308} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1308,\ (\ :0),\ 0.985 + 0.172i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8558341562\)
\(L(\frac12)\) \(\approx\) \(0.8558341562\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.835 + 0.549i)T \)
109 \( 1 + (-0.173 - 0.984i)T \)
good5 \( 1 + (-0.973 + 0.230i)T^{2} \)
7 \( 1 + (0.0694 + 0.0932i)T + (-0.286 + 0.957i)T^{2} \)
11 \( 1 + (-0.396 + 0.918i)T^{2} \)
13 \( 1 + (-0.914 - 1.22i)T + (-0.286 + 0.957i)T^{2} \)
17 \( 1 + (-0.766 + 0.642i)T^{2} \)
19 \( 1 + (0.0201 - 0.114i)T + (-0.939 - 0.342i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.597 + 0.802i)T^{2} \)
31 \( 1 + (-0.914 + 1.22i)T + (-0.286 - 0.957i)T^{2} \)
37 \( 1 + (0.786 + 0.0919i)T + (0.973 + 0.230i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-1.49 - 1.25i)T + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.893 + 0.448i)T^{2} \)
53 \( 1 + (-0.973 + 0.230i)T^{2} \)
59 \( 1 + (-0.597 - 0.802i)T^{2} \)
61 \( 1 + (-0.770 - 0.182i)T + (0.893 + 0.448i)T^{2} \)
67 \( 1 + (0.597 - 0.802i)T + (-0.286 - 0.957i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (-0.479 + 0.315i)T + (0.396 - 0.918i)T^{2} \)
79 \( 1 + (0.227 + 0.526i)T + (-0.686 + 0.727i)T^{2} \)
83 \( 1 + (0.993 + 0.116i)T^{2} \)
89 \( 1 + (0.835 - 0.549i)T^{2} \)
97 \( 1 + (0.744 - 1.72i)T + (-0.686 - 0.727i)T^{2} \)
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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

−9.919514348760781733458638664135, −8.965761545520394519374379465706, −8.141382146493648053374107700170, −7.19033017510770278440139719401, −6.48969550990502809137455891934, −5.84911486970790208384948497027, −4.75662617549301623611104782416, −3.94548314584634265354086641926, −2.42275712637185495850346255579, −1.19426086453071379220434390573, 1.06485833244991883627514993836, 2.94283070365047347932831827139, 3.84872943654671612011891166291, 4.93259247348177022883311913494, 5.62296366072958195534774902611, 6.42266074611269808477582146416, 7.27843156874257388647915375848, 8.423242245006635891937001700420, 9.058891722318540696164757058140, 10.08627004093487869520316389013

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