Properties

Degree 2
Conductor $ 2^{2} \cdot 7 \cdot 11 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.44·3-s + 2·5-s − 7-s + 2.99·9-s − 11-s − 0.449·13-s + 4.89·15-s − 0.449·17-s − 4.89·19-s − 2.44·21-s − 0.898·23-s − 25-s + 2.89·29-s + 6.44·31-s − 2.44·33-s − 2·35-s + 4·37-s − 1.10·39-s − 9.34·41-s + 2.89·43-s + 5.99·45-s − 6.44·47-s + 49-s − 1.10·51-s + 9.79·53-s − 2·55-s − 11.9·57-s + ⋯
L(s)  = 1  + 1.41·3-s + 0.894·5-s − 0.377·7-s + 0.999·9-s − 0.301·11-s − 0.124·13-s + 1.26·15-s − 0.109·17-s − 1.12·19-s − 0.534·21-s − 0.187·23-s − 0.200·25-s + 0.538·29-s + 1.15·31-s − 0.426·33-s − 0.338·35-s + 0.657·37-s − 0.176·39-s − 1.45·41-s + 0.442·43-s + 0.894·45-s − 0.940·47-s + 0.142·49-s − 0.154·51-s + 1.34·53-s − 0.269·55-s − 1.58·57-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{308} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 308,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.09081$
$L(\frac12)$  $\approx$  $2.09081$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
11 \( 1 + T \)
good3 \( 1 - 2.44T + 3T^{2} \)
5 \( 1 - 2T + 5T^{2} \)
13 \( 1 + 0.449T + 13T^{2} \)
17 \( 1 + 0.449T + 17T^{2} \)
19 \( 1 + 4.89T + 19T^{2} \)
23 \( 1 + 0.898T + 23T^{2} \)
29 \( 1 - 2.89T + 29T^{2} \)
31 \( 1 - 6.44T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 9.34T + 41T^{2} \)
43 \( 1 - 2.89T + 43T^{2} \)
47 \( 1 + 6.44T + 47T^{2} \)
53 \( 1 - 9.79T + 53T^{2} \)
59 \( 1 - 7.34T + 59T^{2} \)
61 \( 1 - 5.34T + 61T^{2} \)
67 \( 1 + 14.6T + 67T^{2} \)
71 \( 1 + 12.8T + 71T^{2} \)
73 \( 1 - 12.4T + 73T^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 + 16.8T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 5.10T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.79862544556652117258581879534, −10.32595941868223752647292328970, −9.807840157894385176075972635581, −8.803488770478684976386538125621, −8.159319437625624284725585915287, −6.92091491351518040446086260932, −5.84969430511133985288320968834, −4.34243817203963430244828267007, −2.99018439790270799030868777916, −2.04106593022041204613406247911, 2.04106593022041204613406247911, 2.99018439790270799030868777916, 4.34243817203963430244828267007, 5.84969430511133985288320968834, 6.92091491351518040446086260932, 8.159319437625624284725585915287, 8.803488770478684976386538125621, 9.807840157894385176075972635581, 10.32595941868223752647292328970, 11.79862544556652117258581879534

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