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Properties

Label 2-1032-1032.371-c0-0-0
Degree $2$
Conductor $1032$
Sign $0.458 - 0.888i$
Analytic cond. $0.515035$
Root an. cond. $0.717659$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.222 − 0.974i)2-s + (0.222 + 0.974i)3-s + (−0.900 + 0.433i)4-s + (0.900 − 0.433i)6-s + (0.623 + 0.781i)8-s + (−0.900 + 0.433i)9-s + (−0.846 + 1.75i)11-s + (−0.623 − 0.781i)12-s + (0.623 − 0.781i)16-s + (−0.678 − 0.541i)17-s + (0.623 + 0.781i)18-s + (0.376 + 0.781i)19-s + (1.90 + 0.433i)22-s + (−0.623 + 0.781i)24-s + (−0.222 + 0.974i)25-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (0.222 + 0.974i)3-s + (−0.900 + 0.433i)4-s + (0.900 − 0.433i)6-s + (0.623 + 0.781i)8-s + (−0.900 + 0.433i)9-s + (−0.846 + 1.75i)11-s + (−0.623 − 0.781i)12-s + (0.623 − 0.781i)16-s + (−0.678 − 0.541i)17-s + (0.623 + 0.781i)18-s + (0.376 + 0.781i)19-s + (1.90 + 0.433i)22-s + (−0.623 + 0.781i)24-s + (−0.222 + 0.974i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1032\)    =    \(2^{3} \cdot 3 \cdot 43\)
Sign: $0.458 - 0.888i$
Analytic conductor: \(0.515035\)
Root analytic conductor: \(0.717659\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1032} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1032,\ (\ :0),\ 0.458 - 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7236049804\)
\(L(\frac12)\) \(\approx\) \(0.7236049804\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.222 + 0.974i)T \)
3 \( 1 + (-0.222 - 0.974i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
good5 \( 1 + (0.222 - 0.974i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (0.846 - 1.75i)T + (-0.623 - 0.781i)T^{2} \)
13 \( 1 + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (0.678 + 0.541i)T + (0.222 + 0.974i)T^{2} \)
19 \( 1 + (-0.376 - 0.781i)T + (-0.623 + 0.781i)T^{2} \)
23 \( 1 + (0.623 + 0.781i)T^{2} \)
29 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (0.900 - 0.433i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-1.52 + 0.347i)T + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (0.623 - 0.781i)T^{2} \)
53 \( 1 + (-0.222 - 0.974i)T^{2} \)
59 \( 1 + (-1.52 - 1.21i)T + (0.222 + 0.974i)T^{2} \)
61 \( 1 + (-0.900 - 0.433i)T^{2} \)
67 \( 1 + (-1.80 + 0.867i)T + (0.623 - 0.781i)T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (0.222 - 0.974i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (1.90 + 0.433i)T + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.277 + 1.21i)T + (-0.900 - 0.433i)T^{2} \)
97 \( 1 + (0.400 + 0.193i)T + (0.623 + 0.781i)T^{2} \)
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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

−10.06255839627028207195231002271, −9.739195432769842658308827047880, −8.990145152821753564642273040182, −7.980916137590000807677473383014, −7.27787983207685457910554441688, −5.56872506800682313843033253716, −4.76112673816464866143424879186, −4.08199971345956873925503613199, −2.93014581774759176988059150641, −2.00552000476695780850184317303, 0.71027303754854927997856788619, 2.50963545228650022668822852925, 3.75815495996373369189975912752, 5.17327599026101054382925913458, 5.96868260799295179570119997608, 6.61014690145896450058453425822, 7.53865475268469011225255330976, 8.375246232920054217936101233283, 8.645548679554119015281520359451, 9.733204993555613134951521996641

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