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Properties

Label 2-860-860.619-c0-0-0
Degree $2$
Conductor $860$
Sign $-0.821 + 0.570i$
Analytic cond. $0.429195$
Root an. cond. $0.655130$
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 + (−1.40 + 0.432i)3-s + (−0.900 + 0.433i)4-s + (0.365 − 0.930i)5-s + (0.733 + 1.26i)6-s + (0.222 − 0.385i)7-s + (0.623 + 0.781i)8-s + (0.949 − 0.647i)9-s + (−0.988 − 0.149i)10-s + (1.07 − 0.997i)12-s + (−0.425 − 0.131i)14-s + (−0.109 + 1.46i)15-s + (0.623 − 0.781i)16-s + (−0.842 − 0.781i)18-s + (0.0747 + 0.997i)20-s + (−0.145 + 0.636i)21-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (−1.40 + 0.432i)3-s + (−0.900 + 0.433i)4-s + (0.365 − 0.930i)5-s + (0.733 + 1.26i)6-s + (0.222 − 0.385i)7-s + (0.623 + 0.781i)8-s + (0.949 − 0.647i)9-s + (−0.988 − 0.149i)10-s + (1.07 − 0.997i)12-s + (−0.425 − 0.131i)14-s + (−0.109 + 1.46i)15-s + (0.623 − 0.781i)16-s + (−0.842 − 0.781i)18-s + (0.0747 + 0.997i)20-s + (−0.145 + 0.636i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(860\)    =    \(2^{2} \cdot 5 \cdot 43\)
Sign: $-0.821 + 0.570i$
Analytic conductor: \(0.429195\)
Root analytic conductor: \(0.655130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{860} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 860,\ (\ :0),\ -0.821 + 0.570i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4442813839\)
\(L(\frac12)\) \(\approx\) \(0.4442813839\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.222 + 0.974i)T \)
5 \( 1 + (-0.365 + 0.930i)T \)
43 \( 1 + (0.988 + 0.149i)T \)
good3 \( 1 + (1.40 - 0.432i)T + (0.826 - 0.563i)T^{2} \)
7 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.623 - 0.781i)T^{2} \)
13 \( 1 + (-0.955 + 0.294i)T^{2} \)
17 \( 1 + (0.733 - 0.680i)T^{2} \)
19 \( 1 + (-0.365 - 0.930i)T^{2} \)
23 \( 1 + (0.134 + 1.79i)T + (-0.988 + 0.149i)T^{2} \)
29 \( 1 + (1.88 + 0.582i)T + (0.826 + 0.563i)T^{2} \)
31 \( 1 + (-0.0747 + 0.997i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.162 + 0.712i)T + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.900 + 0.433i)T + (0.623 - 0.781i)T^{2} \)
53 \( 1 + (-0.955 - 0.294i)T^{2} \)
59 \( 1 + (0.222 + 0.974i)T^{2} \)
61 \( 1 + (0.914 + 0.848i)T + (0.0747 + 0.997i)T^{2} \)
67 \( 1 + (-0.603 - 0.411i)T + (0.365 + 0.930i)T^{2} \)
71 \( 1 + (0.988 + 0.149i)T^{2} \)
73 \( 1 + (-0.955 + 0.294i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-1.82 + 0.563i)T + (0.826 - 0.563i)T^{2} \)
89 \( 1 + (-1.57 + 0.487i)T + (0.826 - 0.563i)T^{2} \)
97 \( 1 + (-0.623 - 0.781i)T^{2} \)
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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

−10.32721853603541815979969437909, −9.440649705051125040909546938698, −8.667716336587555663802058812520, −7.64443002966045390595864141531, −6.26482572853964148139889113408, −5.34039849934580634458787186326, −4.66246247525895264307242955885, −3.88835244607670290706099288375, −2.05989637746259023555161580680, −0.59728879185810081389418717149, 1.65248614080162205480690007235, 3.61632512735925350678203945908, 5.11731774246024927336268373523, 5.66709683184244155894519207226, 6.34197607336917666102809542302, 7.17226957172956258352465359679, 7.74199255458778669421324106958, 9.092467486340526679079348791827, 9.848607942151411800321735896169, 10.78502211450979077937157031678

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