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Properties

Label 2-751-751.750-c0-0-6
Degree $2$
Conductor $751$
Sign $1$
Analytic cond. $0.374797$
Root an. cond. $0.612207$
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  + 0.618·2-s − 0.618·4-s + 0.618·5-s − 8-s + 9-s + 0.381·10-s + 0.618·13-s + 0.618·18-s + 2·19-s − 0.381·20-s − 1.61·23-s − 0.618·25-s + 0.381·26-s + 0.999·32-s − 0.618·36-s − 1.61·37-s + 1.23·38-s − 0.618·40-s − 1.61·43-s + 0.618·45-s − 1.00·46-s − 1.61·47-s + 49-s − 0.381·50-s − 0.381·52-s + 2·53-s + 0.618·59-s + ⋯
L(s)  = 1  + 0.618·2-s − 0.618·4-s + 0.618·5-s − 8-s + 9-s + 0.381·10-s + 0.618·13-s + 0.618·18-s + 2·19-s − 0.381·20-s − 1.61·23-s − 0.618·25-s + 0.381·26-s + 0.999·32-s − 0.618·36-s − 1.61·37-s + 1.23·38-s − 0.618·40-s − 1.61·43-s + 0.618·45-s − 1.00·46-s − 1.61·47-s + 49-s − 0.381·50-s − 0.381·52-s + 2·53-s + 0.618·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(751\)
Sign: $1$
Analytic conductor: \(0.374797\)
Root analytic conductor: \(0.612207\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{751} (750, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 751,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.260068016\)
\(L(\frac12)\) \(\approx\) \(1.260068016\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad751 \( 1 - T \)
good2 \( 1 - 0.618T + T^{2} \)
3 \( 1 - T^{2} \)
5 \( 1 - 0.618T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 0.618T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 2T + T^{2} \)
23 \( 1 + 1.61T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 1.61T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.61T + T^{2} \)
47 \( 1 + 1.61T + T^{2} \)
53 \( 1 - 2T + T^{2} \)
59 \( 1 - 0.618T + T^{2} \)
61 \( 1 + 1.61T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 1.61T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.61T + T^{2} \)
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

−10.19777847720186897899466940478, −9.882782118618138297202772899221, −8.974880295137875809079259828672, −8.002465077500857105774712968433, −6.94886816626413160974038483293, −5.87323521664011456151308278376, −5.22580321774063652306187531002, −4.13183066431942222109978492250, −3.29896849958583417787187178261, −1.61869846771177681015428533110, 1.61869846771177681015428533110, 3.29896849958583417787187178261, 4.13183066431942222109978492250, 5.22580321774063652306187531002, 5.87323521664011456151308278376, 6.94886816626413160974038483293, 8.002465077500857105774712968433, 8.974880295137875809079259828672, 9.882782118618138297202772899221, 10.19777847720186897899466940478

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