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Properties

Label 2-2852-2852.275-c0-0-4
Degree $2$
Conductor $2852$
Sign $-0.933 + 0.358i$
Analytic cond. $1.42333$
Root an. cond. $1.19303$
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  + (0.809 − 0.587i)2-s + (0.5 − 1.53i)3-s + (0.309 − 0.951i)4-s + (−0.5 − 1.53i)6-s + (−0.309 − 0.951i)8-s + (−1.30 − 0.951i)9-s + (−1.30 − 0.951i)12-s + (1.11 + 0.363i)13-s + (−0.809 − 0.587i)16-s − 1.61·18-s + (−0.809 − 0.587i)23-s − 1.61·24-s + 25-s + (1.11 − 0.363i)26-s + (−0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.5 − 1.53i)3-s + (0.309 − 0.951i)4-s + (−0.5 − 1.53i)6-s + (−0.309 − 0.951i)8-s + (−1.30 − 0.951i)9-s + (−1.30 − 0.951i)12-s + (1.11 + 0.363i)13-s + (−0.809 − 0.587i)16-s − 1.61·18-s + (−0.809 − 0.587i)23-s − 1.61·24-s + 25-s + (1.11 − 0.363i)26-s + (−0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2852 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2852 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2852\)    =    \(2^{2} \cdot 23 \cdot 31\)
Sign: $-0.933 + 0.358i$
Analytic conductor: \(1.42333\)
Root analytic conductor: \(1.19303\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2852} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2852,\ (\ :0),\ -0.933 + 0.358i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.284512672\)
\(L(\frac12)\) \(\approx\) \(2.284512672\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
good3 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + (0.309 - 0.951i)T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
29 \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + (0.809 - 0.587i)T^{2} \)
47 \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.690 + 0.951i)T + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T^{2} \)
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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

−8.525587176433627845310456896194, −7.85988281533661391686008102841, −6.78054739163088831393006810240, −6.52816051095084037394477924462, −5.68554306035855436076863941082, −4.64252103120474815999576684151, −3.58436526488992492865761200169, −2.83035110850856951746829103349, −1.85859919777739855388983467470, −1.13297535207052476168795798364, 2.22266692067722078629004942571, 3.38000926665113290653826248251, 3.74829625463919097405686323699, 4.53841035093909913931534621015, 5.37475191595444717471015906999, 5.94790061000942500476916924589, 6.93651389451476499701920604572, 8.007265418509914754141169343998, 8.429274705589306830183241988468, 9.255111001781815726053310504243

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