Properties

Label 2-311696-1.1-c1-0-25
Degree $2$
Conductor $311696$
Sign $-1$
Analytic cond. $2488.90$
Root an. cond. $49.8889$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 7-s + 9-s − 6·17-s − 6·19-s − 2·21-s + 23-s − 5·25-s + 4·27-s − 10·29-s − 4·31-s − 2·37-s + 10·41-s − 4·43-s − 12·47-s + 49-s + 12·51-s − 6·53-s + 12·57-s + 2·59-s + 63-s − 2·69-s + 8·71-s + 6·73-s + 10·75-s − 8·79-s − 11·81-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.45·17-s − 1.37·19-s − 0.436·21-s + 0.208·23-s − 25-s + 0.769·27-s − 1.85·29-s − 0.718·31-s − 0.328·37-s + 1.56·41-s − 0.609·43-s − 1.75·47-s + 1/7·49-s + 1.68·51-s − 0.824·53-s + 1.58·57-s + 0.260·59-s + 0.125·63-s − 0.240·69-s + 0.949·71-s + 0.702·73-s + 1.15·75-s − 0.900·79-s − 1.22·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 311696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(311696\)    =    \(2^{4} \cdot 7 \cdot 11^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(2488.90\)
Root analytic conductor: \(49.8889\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{311696} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 311696,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
23 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.79602517209440, −12.57913879600516, −11.77873149756398, −11.27362915570696, −11.19437477269056, −10.89228761066217, −10.27189629559294, −9.662048897837781, −9.332968827533829, −8.584601427885773, −8.430364263635081, −7.738443536355170, −7.122232210687389, −6.819106816051635, −6.183320412033558, −5.863626372532895, −5.447041353179197, −4.714714204447164, −4.510488942043762, −3.888292147667142, −3.329831498737960, −2.443485812704484, −1.957228468987961, −1.518039652185260, −0.4525920594322009, 0, 0.4525920594322009, 1.518039652185260, 1.957228468987961, 2.443485812704484, 3.329831498737960, 3.888292147667142, 4.510488942043762, 4.714714204447164, 5.447041353179197, 5.863626372532895, 6.183320412033558, 6.819106816051635, 7.122232210687389, 7.738443536355170, 8.430364263635081, 8.584601427885773, 9.332968827533829, 9.662048897837781, 10.27189629559294, 10.89228761066217, 11.19437477269056, 11.27362915570696, 11.77873149756398, 12.57913879600516, 12.79602517209440

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