Properties

Label 2-68-1.1-c3-0-3
Degree $2$
Conductor $68$
Sign $-1$
Analytic cond. $4.01212$
Root an. cond. $2.00303$
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  − 2·3-s − 8·5-s − 12·7-s − 23·9-s − 10·11-s − 38·13-s + 16·15-s − 17·17-s + 4·19-s + 24·21-s + 120·23-s − 61·25-s + 100·27-s + 56·29-s + 164·31-s + 20·33-s + 96·35-s − 236·37-s + 76·39-s + 70·41-s − 144·43-s + 184·45-s + 48·47-s − 199·49-s + 34·51-s − 366·53-s + 80·55-s + ⋯
L(s)  = 1  − 0.384·3-s − 0.715·5-s − 0.647·7-s − 0.851·9-s − 0.274·11-s − 0.810·13-s + 0.275·15-s − 0.242·17-s + 0.0482·19-s + 0.249·21-s + 1.08·23-s − 0.487·25-s + 0.712·27-s + 0.358·29-s + 0.950·31-s + 0.105·33-s + 0.463·35-s − 1.04·37-s + 0.312·39-s + 0.266·41-s − 0.510·43-s + 0.609·45-s + 0.148·47-s − 0.580·49-s + 0.0933·51-s − 0.948·53-s + 0.196·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(4.01212\)
Root analytic conductor: \(2.00303\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 68,\ (\ :3/2),\ -1)\)

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 \( 1 \)
17 \( 1 + p T \)
good3 \( 1 + 2 T + p^{3} T^{2} \)
5 \( 1 + 8 T + p^{3} T^{2} \)
7 \( 1 + 12 T + p^{3} T^{2} \)
11 \( 1 + 10 T + p^{3} T^{2} \)
13 \( 1 + 38 T + p^{3} T^{2} \)
19 \( 1 - 4 T + p^{3} T^{2} \)
23 \( 1 - 120 T + p^{3} T^{2} \)
29 \( 1 - 56 T + p^{3} T^{2} \)
31 \( 1 - 164 T + p^{3} T^{2} \)
37 \( 1 + 236 T + p^{3} T^{2} \)
41 \( 1 - 70 T + p^{3} T^{2} \)
43 \( 1 + 144 T + p^{3} T^{2} \)
47 \( 1 - 48 T + p^{3} T^{2} \)
53 \( 1 + 366 T + p^{3} T^{2} \)
59 \( 1 + 504 T + p^{3} T^{2} \)
61 \( 1 + 460 T + p^{3} T^{2} \)
67 \( 1 + 768 T + p^{3} T^{2} \)
71 \( 1 - 72 T + p^{3} T^{2} \)
73 \( 1 + 734 T + p^{3} T^{2} \)
79 \( 1 - 736 T + p^{3} T^{2} \)
83 \( 1 - 856 T + p^{3} T^{2} \)
89 \( 1 - 906 T + p^{3} T^{2} \)
97 \( 1 - 46 T + p^{3} T^{2} \)
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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

−13.69759132405470331436834404289, −12.42906535419894045385995857297, −11.59750746944204852666058138513, −10.44855456406573322489970366974, −9.072712941904369076190514207842, −7.74303749569626472438814751310, −6.39597931959487403216667305779, −4.88936064973963707845787958921, −3.06392505322123889048116412577, 0, 3.06392505322123889048116412577, 4.88936064973963707845787958921, 6.39597931959487403216667305779, 7.74303749569626472438814751310, 9.072712941904369076190514207842, 10.44855456406573322489970366974, 11.59750746944204852666058138513, 12.42906535419894045385995857297, 13.69759132405470331436834404289

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