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Properties

Label 2-670-67.64-c1-0-8
Degree $2$
Conductor $670$
Sign $0.896 - 0.442i$
Analytic cond. $5.34997$
Root an. cond. $2.31300$
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  + (−0.959 − 0.281i)2-s + (−0.325 + 2.26i)3-s + (0.841 + 0.540i)4-s + (−0.415 − 0.909i)5-s + (0.951 − 2.08i)6-s + (3.43 + 1.00i)7-s + (−0.654 − 0.755i)8-s + (−2.15 − 0.632i)9-s + (0.142 + 0.989i)10-s + (−0.744 − 1.62i)11-s + (−1.49 + 1.73i)12-s + (4.18 − 4.83i)13-s + (−3.00 − 1.93i)14-s + (2.19 − 0.645i)15-s + (0.415 + 0.909i)16-s + (1.85 − 1.19i)17-s + ⋯
L(s)  = 1  + (−0.678 − 0.199i)2-s + (−0.188 + 1.30i)3-s + (0.420 + 0.270i)4-s + (−0.185 − 0.406i)5-s + (0.388 − 0.850i)6-s + (1.29 + 0.380i)7-s + (−0.231 − 0.267i)8-s + (−0.718 − 0.210i)9-s + (0.0450 + 0.313i)10-s + (−0.224 − 0.491i)11-s + (−0.433 + 0.499i)12-s + (1.16 − 1.33i)13-s + (−0.804 − 0.516i)14-s + (0.567 − 0.166i)15-s + (0.103 + 0.227i)16-s + (0.449 − 0.288i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.896 - 0.442i$
Analytic conductor: \(5.34997\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{670} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ 0.896 - 0.442i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19160 + 0.278115i\)
\(L(\frac12)\) \(\approx\) \(1.19160 + 0.278115i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.959 + 0.281i)T \)
5 \( 1 + (0.415 + 0.909i)T \)
67 \( 1 + (-6.37 + 5.13i)T \)
good3 \( 1 + (0.325 - 2.26i)T + (-2.87 - 0.845i)T^{2} \)
7 \( 1 + (-3.43 - 1.00i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (0.744 + 1.62i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (-4.18 + 4.83i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (-1.85 + 1.19i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (-4.61 + 1.35i)T + (15.9 - 10.2i)T^{2} \)
23 \( 1 + (-0.588 + 4.09i)T + (-22.0 - 6.47i)T^{2} \)
29 \( 1 + 1.95T + 29T^{2} \)
31 \( 1 + (-0.179 - 0.206i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + 3.50T + 37T^{2} \)
41 \( 1 + (6.73 - 4.32i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (-8.62 + 5.54i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 + (0.409 - 2.84i)T + (-45.0 - 13.2i)T^{2} \)
53 \( 1 + (-2.78 - 1.78i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (-7.23 - 8.35i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (3.26 - 7.15i)T + (-39.9 - 46.1i)T^{2} \)
71 \( 1 + (-9.21 - 5.92i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (3.40 - 7.45i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (10.6 - 12.3i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (4.64 + 10.1i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.454 + 3.16i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 - 5.62T + 97T^{2} \)
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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

−10.58883468275199828249628922200, −9.816813038774553927387003131667, −8.736529053591821916439015271101, −8.379989950613634678923793919872, −7.40852781319969155030166179122, −5.66462911268600727989597513492, −5.18969460904801464684065073026, −4.02075960114258276545449146294, −2.94622028795460749047767440833, −1.08763668207186419312400986240, 1.28312917214059056303440790896, 1.93755688609328590377210220651, 3.78660553173754669625791494983, 5.24089483190539547567635260266, 6.33507423795508987361860328443, 7.16324745899187468935284016105, 7.69341465390676839570198943692, 8.396223442308548036270498974594, 9.481899459915330608431696329818, 10.57618945617190791648475167154

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