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Properties

Label 2-680-85.73-c1-0-21
Degree $2$
Conductor $680$
Sign $-0.579 + 0.814i$
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  + (−0.446 + 2.24i)3-s + (0.682 − 2.12i)5-s + (−1.62 − 1.08i)7-s + (−2.06 − 0.856i)9-s + (−3.24 − 2.17i)11-s − 0.242i·13-s + (4.47 + 2.48i)15-s + (−4.12 + 0.0694i)17-s + (−6.48 + 2.68i)19-s + (3.16 − 3.16i)21-s + (−9.03 + 1.79i)23-s + (−4.06 − 2.90i)25-s + (−0.968 + 1.44i)27-s + (1.52 − 7.66i)29-s + (4.24 − 2.83i)31-s + ⋯
L(s)  = 1  + (−0.257 + 1.29i)3-s + (0.305 − 0.952i)5-s + (−0.614 − 0.410i)7-s + (−0.689 − 0.285i)9-s + (−0.979 − 0.654i)11-s − 0.0672i·13-s + (1.15 + 0.640i)15-s + (−0.999 + 0.0168i)17-s + (−1.48 + 0.616i)19-s + (0.690 − 0.690i)21-s + (−1.88 + 0.374i)23-s + (−0.813 − 0.580i)25-s + (−0.186 + 0.278i)27-s + (0.283 − 1.42i)29-s + (0.762 − 0.509i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.579 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.579 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $-0.579 + 0.814i$
Analytic conductor: \(5.42982\)
Root analytic conductor: \(2.33019\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 680,\ (\ :1/2),\ -0.579 + 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.127538 - 0.247304i\)
\(L(\frac12)\) \(\approx\) \(0.127538 - 0.247304i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.682 + 2.12i)T \)
17 \( 1 + (4.12 - 0.0694i)T \)
good3 \( 1 + (0.446 - 2.24i)T + (-2.77 - 1.14i)T^{2} \)
7 \( 1 + (1.62 + 1.08i)T + (2.67 + 6.46i)T^{2} \)
11 \( 1 + (3.24 + 2.17i)T + (4.20 + 10.1i)T^{2} \)
13 \( 1 + 0.242iT - 13T^{2} \)
19 \( 1 + (6.48 - 2.68i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (9.03 - 1.79i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (-1.52 + 7.66i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + (-4.24 + 2.83i)T + (11.8 - 28.6i)T^{2} \)
37 \( 1 + (-2.25 - 0.447i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (-0.789 - 3.96i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (-9.56 + 3.96i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 7.19T + 47T^{2} \)
53 \( 1 + (4.13 - 9.97i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-2.44 + 5.89i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-4.60 + 0.915i)T + (56.3 - 23.3i)T^{2} \)
67 \( 1 + (-3.29 - 3.29i)T + 67iT^{2} \)
71 \( 1 + (5.28 + 7.90i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (10.4 - 6.97i)T + (27.9 - 67.4i)T^{2} \)
79 \( 1 + (5.04 - 7.54i)T + (-30.2 - 72.9i)T^{2} \)
83 \( 1 + (-2.46 - 1.02i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (4.74 + 4.74i)T + 89iT^{2} \)
97 \( 1 + (-1.83 + 1.22i)T + (37.1 - 89.6i)T^{2} \)
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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

−10.13918452894762812599976072854, −9.570124976895306693450063661967, −8.577923560657290867248693545980, −7.86903628503615796257868965696, −6.18819319467935899641469031100, −5.67516673376522628314182648903, −4.34973665915737383951073732445, −4.07311449675191773578054249107, −2.38056627137422767556279793882, −0.13729965291871746544784370538, 2.08116200123129841410594603467, 2.64633197357120025285735991280, 4.33614436921067075772801522894, 5.82463561841383018457071713924, 6.49576627730760336598067372655, 7.06623924095320978854612423049, 7.963928303634515191453447021257, 8.978669084658807293612291434467, 10.15658061333543753106091690092, 10.71791232016747136509089565745

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