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Properties

Label 2-2852-2852.275-c0-0-5
Degree $2$
Conductor $2852$
Sign $-0.629 - 0.777i$
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  + (0.5 − 0.866i)2-s + (0.564 − 1.73i)3-s + (−0.499 − 0.866i)4-s + (−1.22 − 1.35i)6-s − 0.999·8-s + (−1.89 − 1.37i)9-s + (−1.78 + 0.379i)12-s + (−1.89 − 0.614i)13-s + (−0.5 + 0.866i)16-s + (−2.13 + 0.951i)18-s + (0.809 + 0.587i)23-s + (−0.564 + 1.73i)24-s + 25-s + (−1.47 + 1.33i)26-s + (−1.97 + 1.43i)27-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.564 − 1.73i)3-s + (−0.499 − 0.866i)4-s + (−1.22 − 1.35i)6-s − 0.999·8-s + (−1.89 − 1.37i)9-s + (−1.78 + 0.379i)12-s + (−1.89 − 0.614i)13-s + (−0.5 + 0.866i)16-s + (−2.13 + 0.951i)18-s + (0.809 + 0.587i)23-s + (−0.564 + 1.73i)24-s + 25-s + (−1.47 + 1.33i)26-s + (−1.97 + 1.43i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2852 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2852 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2852\)    =    \(2^{2} \cdot 23 \cdot 31\)
Sign: $-0.629 - 0.777i$
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: \((2,\ 2852,\ (\ :0),\ -0.629 - 0.777i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.423604008\)
\(L(\frac12)\) \(\approx\) \(1.423604008\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
good3 \( 1 + (-0.564 + 1.73i)T + (-0.809 - 0.587i)T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + (0.309 - 0.951i)T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (1.89 + 0.614i)T + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
29 \( 1 + (-1.89 + 0.614i)T + (0.809 - 0.587i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.564 + 1.73i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + (0.809 - 0.587i)T^{2} \)
47 \( 1 + (0.395 + 0.128i)T + (0.809 + 0.587i)T^{2} \)
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 \( 1 + (0.478 - 0.658i)T + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (1.16 + 1.60i)T + (-0.309 + 0.951i)T^{2} \)
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

−8.473853102245361387347955346377, −7.67435763796304811676225383793, −6.97109113384735054398494642271, −6.30977465528872978315536220163, −5.32061190472111420932131954396, −4.57960739033309194792994546301, −2.98506825025111530656296867061, −2.81894985080836979277876818152, −1.77293987479350937765913470416, −0.69248685122205380875314174086, 2.75681033411097617133817397650, 3.02863288259921570218304690108, 4.36904766831738294884924506769, 4.72148092674370181405542350795, 5.16135969665509250054256576627, 6.41348185692001339652821793914, 7.11389117541066040845962210781, 8.162724376923925430896855411719, 8.637902802428439863415522034906, 9.404929032441528033586131418713

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