Properties

Label 2-338-1.1-c3-0-37
Degree $2$
Conductor $338$
Sign $-1$
Analytic cond. $19.9426$
Root an. cond. $4.46571$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·3-s + 4·4-s − 11·5-s + 6·6-s − 19·7-s + 8·8-s − 18·9-s − 22·10-s + 38·11-s + 12·12-s − 38·14-s − 33·15-s + 16·16-s − 51·17-s − 36·18-s − 90·19-s − 44·20-s − 57·21-s + 76·22-s − 52·23-s + 24·24-s − 4·25-s − 135·27-s − 76·28-s − 190·29-s − 66·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.983·5-s + 0.408·6-s − 1.02·7-s + 0.353·8-s − 2/3·9-s − 0.695·10-s + 1.04·11-s + 0.288·12-s − 0.725·14-s − 0.568·15-s + 1/4·16-s − 0.727·17-s − 0.471·18-s − 1.08·19-s − 0.491·20-s − 0.592·21-s + 0.736·22-s − 0.471·23-s + 0.204·24-s − 0.0319·25-s − 0.962·27-s − 0.512·28-s − 1.21·29-s − 0.401·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(19.9426\)
Root analytic conductor: \(4.46571\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 338,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
13 \( 1 \)
good3 \( 1 - p T + p^{3} T^{2} \)
5 \( 1 + 11 T + p^{3} T^{2} \)
7 \( 1 + 19 T + p^{3} T^{2} \)
11 \( 1 - 38 T + p^{3} T^{2} \)
17 \( 1 + 3 p T + p^{3} T^{2} \)
19 \( 1 + 90 T + p^{3} T^{2} \)
23 \( 1 + 52 T + p^{3} T^{2} \)
29 \( 1 + 190 T + p^{3} T^{2} \)
31 \( 1 + 292 T + p^{3} T^{2} \)
37 \( 1 - 441 T + p^{3} T^{2} \)
41 \( 1 + 312 T + p^{3} T^{2} \)
43 \( 1 - 373 T + p^{3} T^{2} \)
47 \( 1 - 41 T + p^{3} T^{2} \)
53 \( 1 - 468 T + p^{3} T^{2} \)
59 \( 1 + 530 T + p^{3} T^{2} \)
61 \( 1 - 592 T + p^{3} T^{2} \)
67 \( 1 - 206 T + p^{3} T^{2} \)
71 \( 1 - 863 T + p^{3} T^{2} \)
73 \( 1 - 322 T + p^{3} T^{2} \)
79 \( 1 + 460 T + p^{3} T^{2} \)
83 \( 1 + 528 T + p^{3} T^{2} \)
89 \( 1 + 870 T + p^{3} T^{2} \)
97 \( 1 - 346 T + p^{3} T^{2} \)
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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

−11.02033836973727312194534387419, −9.561707948196710660097303311873, −8.777804510295442758525534294877, −7.71736993876697203653098806665, −6.70145270281234560532829577345, −5.79706258933615279319249186213, −4.08948772314240643530253502513, −3.62148059983975687591165031601, −2.29522510715714479838737487092, 0, 2.29522510715714479838737487092, 3.62148059983975687591165031601, 4.08948772314240643530253502513, 5.79706258933615279319249186213, 6.70145270281234560532829577345, 7.71736993876697203653098806665, 8.777804510295442758525534294877, 9.561707948196710660097303311873, 11.02033836973727312194534387419

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