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Properties

Label 2-2527-133.125-c0-0-10
Degree $2$
Conductor $2527$
Sign $-0.247 - 0.968i$
Analytic cond. $1.26113$
Root an. cond. $1.12300$
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  + (0.951 − 1.64i)2-s + (−1.30 − 2.26i)4-s − 7-s − 3.07·8-s + (−0.5 − 0.866i)9-s − 0.618·11-s + (−0.951 + 1.64i)14-s + (−1.61 + 2.80i)16-s − 1.90·18-s + (−0.587 + 1.01i)22-s + (0.809 + 1.40i)23-s + (−0.5 − 0.866i)25-s + (1.30 + 2.26i)28-s + (−0.587 − 1.01i)29-s + (1.53 + 2.66i)32-s + ⋯
L(s)  = 1  + (0.951 − 1.64i)2-s + (−1.30 − 2.26i)4-s − 7-s − 3.07·8-s + (−0.5 − 0.866i)9-s − 0.618·11-s + (−0.951 + 1.64i)14-s + (−1.61 + 2.80i)16-s − 1.90·18-s + (−0.587 + 1.01i)22-s + (0.809 + 1.40i)23-s + (−0.5 − 0.866i)25-s + (1.30 + 2.26i)28-s + (−0.587 − 1.01i)29-s + (1.53 + 2.66i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2527\)    =    \(7 \cdot 19^{2}\)
Sign: $-0.247 - 0.968i$
Analytic conductor: \(1.26113\)
Root analytic conductor: \(1.12300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2527} (790, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2527,\ (\ :0),\ -0.247 - 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9055595924\)
\(L(\frac12)\) \(\approx\) \(0.9055595924\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + T \)
19 \( 1 \)
good2 \( 1 + (-0.951 + 1.64i)T + (-0.5 - 0.866i)T^{2} \)
3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + 0.618T + T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.587 + 1.01i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.951 + 1.64i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.951 + 1.64i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.587 + 1.01i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−9.115997632919428609510682737466, −7.929046250210582287036724113066, −6.61207547657975115359250048467, −5.93561428490229656218159855518, −5.26709059003049480243806091689, −4.25543128735468337826346722083, −3.39684096170206129168414803904, −2.97522153790312227353177541027, −1.88382577365323910547676609184, −0.40977854478102481313479511475, 2.64044135573148833259680948686, 3.35583274762951241476611576464, 4.37016614037918626415264224007, 5.19253491687372144034145525566, 5.72022792618944342571970164363, 6.52651162897209135950423168715, 7.23012788375196338063223149452, 7.79636418587347703027508655794, 8.682815504807854260118405133177, 9.144170061859238949038749196127

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