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Properties

Label 2-680-680.99-c1-0-56
Degree $2$
Conductor $680$
Sign $0.859 - 0.510i$
Analytic cond. $5.42982$
Root an. cond. $2.33019$
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  + (−1.20 + 0.740i)2-s + (1.63 + 0.324i)3-s + (0.902 − 1.78i)4-s + (2.22 + 0.260i)5-s + (−2.20 + 0.818i)6-s + (−0.693 − 0.463i)7-s + (0.234 + 2.81i)8-s + (−0.211 − 0.0876i)9-s + (−2.86 + 1.33i)10-s + (5.70 − 1.13i)11-s + (2.05 − 2.62i)12-s + (−0.791 − 0.791i)13-s + (1.17 + 0.0445i)14-s + (3.54 + 1.14i)15-s + (−2.37 − 3.22i)16-s + (−1.86 + 3.67i)17-s + ⋯
L(s)  = 1  + (−0.851 + 0.523i)2-s + (0.942 + 0.187i)3-s + (0.451 − 0.892i)4-s + (0.993 + 0.116i)5-s + (−0.901 + 0.333i)6-s + (−0.261 − 0.175i)7-s + (0.0828 + 0.996i)8-s + (−0.0705 − 0.0292i)9-s + (−0.907 + 0.420i)10-s + (1.71 − 0.341i)11-s + (0.592 − 0.756i)12-s + (−0.219 − 0.219i)13-s + (0.314 + 0.0119i)14-s + (0.914 + 0.296i)15-s + (−0.592 − 0.805i)16-s + (−0.451 + 0.892i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.510i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.859 - 0.510i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ 0.859 - 0.510i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61536 + 0.443332i\)
\(L(\frac12)\) \(\approx\) \(1.61536 + 0.443332i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.20 - 0.740i)T \)
5 \( 1 + (-2.22 - 0.260i)T \)
17 \( 1 + (1.86 - 3.67i)T \)
good3 \( 1 + (-1.63 - 0.324i)T + (2.77 + 1.14i)T^{2} \)
7 \( 1 + (0.693 + 0.463i)T + (2.67 + 6.46i)T^{2} \)
11 \( 1 + (-5.70 + 1.13i)T + (10.1 - 4.20i)T^{2} \)
13 \( 1 + (0.791 + 0.791i)T + 13iT^{2} \)
19 \( 1 + (-1.74 + 0.720i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-0.716 - 3.59i)T + (-21.2 + 8.80i)T^{2} \)
29 \( 1 + (-7.13 + 4.76i)T + (11.0 - 26.7i)T^{2} \)
31 \( 1 + (0.420 - 2.11i)T + (-28.6 - 11.8i)T^{2} \)
37 \( 1 + (-0.424 + 2.13i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (3.08 + 2.06i)T + (15.6 + 37.8i)T^{2} \)
43 \( 1 + (1.72 - 4.16i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + (7.04 + 7.04i)T + 47iT^{2} \)
53 \( 1 + (-8.10 + 3.35i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (4.00 - 9.67i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-8.90 - 5.94i)T + (23.3 + 56.3i)T^{2} \)
67 \( 1 + 9.22T + 67T^{2} \)
71 \( 1 + (-2.73 - 0.544i)T + (65.5 + 27.1i)T^{2} \)
73 \( 1 + (-9.15 - 13.7i)T + (-27.9 + 67.4i)T^{2} \)
79 \( 1 + (-0.721 - 3.62i)T + (-72.9 + 30.2i)T^{2} \)
83 \( 1 + (-1.61 + 0.670i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (8.32 - 8.32i)T - 89iT^{2} \)
97 \( 1 + (1.19 - 0.801i)T + (37.1 - 89.6i)T^{2} \)
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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

−10.02607821298375515293908468100, −9.640602552606013053218632347866, −8.780273641618127526743562363883, −8.348348172666878025176054863309, −6.96655044862868406242354479989, −6.37971444687615358374928205441, −5.44426622828805758213250342062, −3.86470862664225355229991081549, −2.60780556521602655375639515633, −1.36851558854920171247821240209, 1.39752670580209258308780512043, 2.43533824221657628128978975379, 3.32450583714437665206915099318, 4.71602453975520073932696258291, 6.38661681966547518771974159081, 6.93762302820591489785796006709, 8.141392069643907974021865544999, 9.059761228330906685470755957494, 9.283602647504945994003237554918, 10.08306045429649632215614095364

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