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Properties

Label 2-968-8.3-c0-0-3
Degree $2$
Conductor $968$
Sign $1$
Analytic cond. $0.483094$
Root an. cond. $0.695050$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 0.618·3-s + 4-s + 0.618·6-s + 8-s − 0.618·9-s + 0.618·12-s + 16-s − 1.61·17-s − 0.618·18-s − 1.61·19-s + 0.618·24-s + 25-s − 27-s + 32-s − 1.61·34-s − 0.618·36-s − 1.61·38-s + 0.618·41-s + 0.618·43-s + 0.618·48-s + 49-s + 50-s − 1.00·51-s − 54-s − 1.00·57-s − 1.61·59-s + ⋯
L(s)  = 1  + 2-s + 0.618·3-s + 4-s + 0.618·6-s + 8-s − 0.618·9-s + 0.618·12-s + 16-s − 1.61·17-s − 0.618·18-s − 1.61·19-s + 0.618·24-s + 25-s − 27-s + 32-s − 1.61·34-s − 0.618·36-s − 1.61·38-s + 0.618·41-s + 0.618·43-s + 0.618·48-s + 49-s + 50-s − 1.00·51-s − 54-s − 1.00·57-s − 1.61·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(0.483094\)
Root analytic conductor: \(0.695050\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (243, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.059817545\)
\(L(\frac12)\) \(\approx\) \(2.059817545\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
11 \( 1 \)
good3 \( 1 - 0.618T + T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 1.61T + T^{2} \)
19 \( 1 + 1.61T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 0.618T + T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.61T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.61T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 0.618T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 0.618T + T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 - 0.618T + T^{2} \)
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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

−10.64385339674351399931644087072, −9.171876591710188894447272275791, −8.581359837593480654930778526542, −7.61651870839918640848470123023, −6.62364498657474985244173047427, −5.97272482646368570277264553751, −4.76376083724081524472618929305, −4.02762265538057813632654740576, −2.84940028320597930542443209980, −2.09388735580038497118463819268, 2.09388735580038497118463819268, 2.84940028320597930542443209980, 4.02762265538057813632654740576, 4.76376083724081524472618929305, 5.97272482646368570277264553751, 6.62364498657474985244173047427, 7.61651870839918640848470123023, 8.581359837593480654930778526542, 9.171876591710188894447272275791, 10.64385339674351399931644087072

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