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Properties

Label 2-260-20.7-c1-0-27
Degree $2$
Conductor $260$
Sign $0.728 + 0.685i$
Analytic cond. $2.07611$
Root an. cond. $1.44087$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 − 0.0447i)2-s + (−1.60 − 1.60i)3-s + (1.99 − 0.126i)4-s + (1.86 + 1.23i)5-s + (−2.34 − 2.20i)6-s + (0.275 − 0.275i)7-s + (2.81 − 0.267i)8-s + 2.17i·9-s + (2.68 + 1.66i)10-s − 0.822i·11-s + (−3.41 − 3.00i)12-s + (0.707 − 0.707i)13-s + (0.377 − 0.402i)14-s + (−1.00 − 4.98i)15-s + (3.96 − 0.504i)16-s + (−4.59 − 4.59i)17-s + ⋯
L(s)  = 1  + (0.999 − 0.0316i)2-s + (−0.928 − 0.928i)3-s + (0.998 − 0.0632i)4-s + (0.832 + 0.553i)5-s + (−0.957 − 0.898i)6-s + (0.104 − 0.104i)7-s + (0.995 − 0.0947i)8-s + 0.724i·9-s + (0.850 + 0.526i)10-s − 0.247i·11-s + (−0.985 − 0.867i)12-s + (0.196 − 0.196i)13-s + (0.100 − 0.107i)14-s + (−0.259 − 1.28i)15-s + (0.992 − 0.126i)16-s + (−1.11 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.728 + 0.685i$
Analytic conductor: \(2.07611\)
Root analytic conductor: \(1.44087\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 260,\ (\ :1/2),\ 0.728 + 0.685i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81275 - 0.718655i\)
\(L(\frac12)\) \(\approx\) \(1.81275 - 0.718655i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0447i)T \)
5 \( 1 + (-1.86 - 1.23i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (1.60 + 1.60i)T + 3iT^{2} \)
7 \( 1 + (-0.275 + 0.275i)T - 7iT^{2} \)
11 \( 1 + 0.822iT - 11T^{2} \)
17 \( 1 + (4.59 + 4.59i)T + 17iT^{2} \)
19 \( 1 + 2.21T + 19T^{2} \)
23 \( 1 + (-2.72 - 2.72i)T + 23iT^{2} \)
29 \( 1 - 7.29iT - 29T^{2} \)
31 \( 1 - 9.35iT - 31T^{2} \)
37 \( 1 + (4.29 + 4.29i)T + 37iT^{2} \)
41 \( 1 - 1.02T + 41T^{2} \)
43 \( 1 + (4.26 + 4.26i)T + 43iT^{2} \)
47 \( 1 + (-3.16 + 3.16i)T - 47iT^{2} \)
53 \( 1 + (-0.615 + 0.615i)T - 53iT^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 - 1.78T + 61T^{2} \)
67 \( 1 + (7.04 - 7.04i)T - 67iT^{2} \)
71 \( 1 + 3.71iT - 71T^{2} \)
73 \( 1 + (-0.126 + 0.126i)T - 73iT^{2} \)
79 \( 1 - 8.97T + 79T^{2} \)
83 \( 1 + (3.04 + 3.04i)T + 83iT^{2} \)
89 \( 1 + 12.5iT - 89T^{2} \)
97 \( 1 + (3.35 + 3.35i)T + 97iT^{2} \)
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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

−12.03019819657889687102441303972, −11.03013087302364409127196541943, −10.60999417568556722829779954810, −9.006645835425643225096924573074, −7.17988295831372226905668127399, −6.80876604625530860317152059492, −5.79831745100024375204688273753, −4.96607724119136644466869663877, −3.11164345451272434465483477066, −1.64481845906104782647487083813, 2.13838249307860764357920684120, 4.16740655129940546211901704464, 4.78398785924724585523879115592, 5.89332667439160838606856889336, 6.46743174501995852626223923313, 8.224661685666675219414581511724, 9.549384685739373992488336734122, 10.49349303656495631301078597023, 11.15155554400295092704602439381, 12.11190677030105156670932039751

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