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Properties

Label 2-3100-620.439-c0-0-1
Degree $2$
Conductor $3100$
Sign $0.334 - 0.942i$
Analytic cond. $1.54710$
Root an. cond. $1.24382$
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  + 2-s + (−0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (0.866 − 0.5i)11-s + (−0.5 + 0.866i)12-s + (0.866 − 0.5i)13-s + (−0.5 + 0.866i)14-s + 16-s + (−0.866 − 0.5i)17-s + (0.866 + 0.5i)19-s + (−0.499 − 0.866i)21-s + (0.866 − 0.5i)22-s + ⋯
L(s)  = 1  + 2-s + (−0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)6-s + (−0.5 + 0.866i)7-s + 8-s + (0.866 − 0.5i)11-s + (−0.5 + 0.866i)12-s + (0.866 − 0.5i)13-s + (−0.5 + 0.866i)14-s + 16-s + (−0.866 − 0.5i)17-s + (0.866 + 0.5i)19-s + (−0.499 − 0.866i)21-s + (0.866 − 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3100\)    =    \(2^{2} \cdot 5^{2} \cdot 31\)
Sign: $0.334 - 0.942i$
Analytic conductor: \(1.54710\)
Root analytic conductor: \(1.24382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3100} (2299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3100,\ (\ :0),\ 0.334 - 0.942i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.145042282\)
\(L(\frac12)\) \(\approx\) \(2.145042282\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 \)
31 \( 1 - iT \)
good3 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
37 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.059201486478345986913352170924, −8.385452844529253053269588592912, −7.22984041900066411618286028149, −6.49654498296735004956818305263, −5.68847091455104255482246287730, −5.34045594637151400816262619322, −4.36133506507771311799667529868, −3.60409136017536410054609815505, −2.91152825212158226549359492547, −1.60398469939576053752727578108, 1.14953641591110374745511598523, 1.97577363131567864816377018114, 3.39232547889014650728683080049, 4.00874107000753962397629509879, 4.78094655509316772459830014847, 5.98849585522427375038919657611, 6.41309620907686691632061984662, 7.04978446595181953375101551858, 7.44762588821200449430172462702, 8.639100339679315018591954244564

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