Properties

Degree 2
Conductor 7
Sign $1$
Motivic weight 16
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 449·2-s + 1.36e5·4-s + 5.76e6·7-s + 3.16e7·8-s + 4.30e7·9-s − 2.55e8·11-s + 2.58e9·14-s + 5.30e9·16-s + 1.93e10·18-s − 1.14e11·22-s − 1.56e11·23-s + 1.52e11·25-s + 7.84e11·28-s − 9.88e11·29-s + 3.05e11·32-s + 5.85e12·36-s − 2.72e12·37-s + 2.18e13·43-s − 3.47e13·44-s − 7.01e13·46-s + 3.32e13·49-s + 6.85e13·50-s + 1.11e14·53-s + 1.82e14·56-s − 4.43e14·58-s + 2.48e14·63-s − 2.10e14·64-s + ⋯
L(s)  = 1  + 1.75·2-s + 2.07·4-s + 7-s + 1.88·8-s + 9-s − 1.19·11-s + 1.75·14-s + 1.23·16-s + 1.75·18-s − 2.09·22-s − 1.99·23-s + 25-s + 2.07·28-s − 1.97·29-s + 0.277·32-s + 2.07·36-s − 0.775·37-s + 1.87·43-s − 2.47·44-s − 3.49·46-s + 49-s + 1.75·50-s + 1.79·53-s + 1.88·56-s − 3.46·58-s + 63-s − 0.747·64-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(16\)
character  :  $\chi_{7} (6, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 7,\ (\ :8),\ 1)$
$L(\frac{17}{2})$  $\approx$  $4.98071$
$L(\frac12)$  $\approx$  $4.98071$
$L(9)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 7$, \(F_p\) is a polynomial of degree 2. If $p = 7$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad7 \( 1 - p^{8} T \)
good2 \( 1 - 449 T + p^{16} T^{2} \)
3 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
5 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
11 \( 1 + 255690046 T + p^{16} T^{2} \)
13 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
17 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
19 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
23 \( 1 + 156184073086 T + p^{16} T^{2} \)
29 \( 1 + 988786884286 T + p^{16} T^{2} \)
31 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
37 \( 1 + 2723955766846 T + p^{16} T^{2} \)
41 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
43 \( 1 - 21863294238914 T + p^{16} T^{2} \)
47 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
53 \( 1 - 111956183305922 T + p^{16} T^{2} \)
59 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
61 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
67 \( 1 + 561251106979006 T + p^{16} T^{2} \)
71 \( 1 - 500488282933634 T + p^{16} T^{2} \)
73 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
79 \( 1 - 1139826930254594 T + p^{16} T^{2} \)
83 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
89 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
97 \( ( 1 - p^{8} T )( 1 + p^{8} T ) \)
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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

−18.30028539990631822834356103685, −16.05041840561314309358051150973, −14.90072619904039714878660886462, −13.55146067813200276718524465476, −12.31247967783446303743988546586, −10.73097059236948430174940892727, −7.49546843852742373270098558066, −5.48094072653103701873658355340, −4.13277098119284596651574053906, −2.08481030511200200867664303185, 2.08481030511200200867664303185, 4.13277098119284596651574053906, 5.48094072653103701873658355340, 7.49546843852742373270098558066, 10.73097059236948430174940892727, 12.31247967783446303743988546586, 13.55146067813200276718524465476, 14.90072619904039714878660886462, 16.05041840561314309358051150973, 18.30028539990631822834356103685

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