Properties

Label 2-338-1.1-c7-0-5
Degree $2$
Conductor $338$
Sign $1$
Analytic cond. $105.586$
Root an. cond. $10.2755$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s − 87·3-s + 64·4-s − 321·5-s + 696·6-s + 181·7-s − 512·8-s + 5.38e3·9-s + 2.56e3·10-s − 7.78e3·11-s − 5.56e3·12-s − 1.44e3·14-s + 2.79e4·15-s + 4.09e3·16-s + 9.06e3·17-s − 4.30e4·18-s + 3.71e4·19-s − 2.05e4·20-s − 1.57e4·21-s + 6.22e4·22-s + 1.90e4·23-s + 4.45e4·24-s + 2.49e4·25-s − 2.77e5·27-s + 1.15e4·28-s + 1.74e5·29-s − 2.23e5·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.86·3-s + 1/2·4-s − 1.14·5-s + 1.31·6-s + 0.199·7-s − 0.353·8-s + 2.46·9-s + 0.812·10-s − 1.76·11-s − 0.930·12-s − 0.141·14-s + 2.13·15-s + 1/4·16-s + 0.447·17-s − 1.74·18-s + 1.24·19-s − 0.574·20-s − 0.371·21-s + 1.24·22-s + 0.325·23-s + 0.657·24-s + 0.318·25-s − 2.71·27-s + 0.0997·28-s + 1.33·29-s − 1.51·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(105.586\)
Root analytic conductor: \(10.2755\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(0.1463776781\)
\(L(\frac12)\) \(\approx\) \(0.1463776781\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{3} T \)
13 \( 1 \)
good3 \( 1 + 29 p T + p^{7} T^{2} \)
5 \( 1 + 321 T + p^{7} T^{2} \)
7 \( 1 - 181 T + p^{7} T^{2} \)
11 \( 1 + 7782 T + p^{7} T^{2} \)
17 \( 1 - 9069 T + p^{7} T^{2} \)
19 \( 1 - 37150 T + p^{7} T^{2} \)
23 \( 1 - 19008 T + p^{7} T^{2} \)
29 \( 1 - 174750 T + p^{7} T^{2} \)
31 \( 1 + 29012 T + p^{7} T^{2} \)
37 \( 1 + 323669 T + p^{7} T^{2} \)
41 \( 1 + 795312 T + p^{7} T^{2} \)
43 \( 1 + 314137 T + p^{7} T^{2} \)
47 \( 1 - 447441 T + p^{7} T^{2} \)
53 \( 1 + 1469232 T + p^{7} T^{2} \)
59 \( 1 + 1627770 T + p^{7} T^{2} \)
61 \( 1 + 2399608 T + p^{7} T^{2} \)
67 \( 1 - 64066 T + p^{7} T^{2} \)
71 \( 1 - 322383 T + p^{7} T^{2} \)
73 \( 1 - 4454782 T + p^{7} T^{2} \)
79 \( 1 - 753560 T + p^{7} T^{2} \)
83 \( 1 - 1219092 T + p^{7} T^{2} \)
89 \( 1 + 3390330 T + p^{7} T^{2} \)
97 \( 1 + 1628774 T + p^{7} T^{2} \)
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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

−10.57251443856064702806177093541, −9.796341965026754185807313752930, −8.137551617548105257730129744132, −7.52775501313748868207606690895, −6.62577424829312118012093098319, −5.36797093315704298529232610977, −4.81232022304106188517015575116, −3.23445219483646376045155670957, −1.36141817643905478023521609211, −0.24329062095540364962349798947, 0.24329062095540364962349798947, 1.36141817643905478023521609211, 3.23445219483646376045155670957, 4.81232022304106188517015575116, 5.36797093315704298529232610977, 6.62577424829312118012093098319, 7.52775501313748868207606690895, 8.137551617548105257730129744132, 9.796341965026754185807313752930, 10.57251443856064702806177093541

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