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Properties

Label 2-670-335.104-c1-0-13
Degree $2$
Conductor $670$
Sign $0.469 - 0.883i$
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  + (0.866 − 0.5i)2-s + 1.76i·3-s + (0.499 − 0.866i)4-s + (0.428 + 2.19i)5-s + (0.882 + 1.52i)6-s + (2.08 + 1.20i)7-s − 0.999i·8-s − 0.111·9-s + (1.46 + 1.68i)10-s + (1.36 − 2.35i)11-s + (1.52 + 0.882i)12-s + (−2.79 + 1.61i)13-s + 2.41·14-s + (−3.87 + 0.755i)15-s + (−0.5 − 0.866i)16-s + (1.94 − 1.12i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + 1.01i·3-s + (0.249 − 0.433i)4-s + (0.191 + 0.981i)5-s + (0.360 + 0.623i)6-s + (0.789 + 0.455i)7-s − 0.353i·8-s − 0.0373·9-s + (0.464 + 0.533i)10-s + (0.410 − 0.710i)11-s + (0.441 + 0.254i)12-s + (−0.774 + 0.446i)13-s + 0.644·14-s + (−0.999 + 0.195i)15-s + (−0.125 − 0.216i)16-s + (0.472 − 0.272i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.469 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $0.469 - 0.883i$
Analytic conductor: \(5.34997\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{670} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ 0.469 - 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03811 + 1.22506i\)
\(L(\frac12)\) \(\approx\) \(2.03811 + 1.22506i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-0.428 - 2.19i)T \)
67 \( 1 + (-5.79 - 5.78i)T \)
good3 \( 1 - 1.76iT - 3T^{2} \)
7 \( 1 + (-2.08 - 1.20i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.36 + 2.35i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.79 - 1.61i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.94 + 1.12i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0706 - 0.122i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.60 - 2.08i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.582 - 1.00i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.242 + 0.420i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.73 + 5.04i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.77 - 6.53i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 3.17iT - 43T^{2} \)
47 \( 1 + (-0.647 - 0.373i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.72iT - 53T^{2} \)
59 \( 1 - 0.299T + 59T^{2} \)
61 \( 1 + (0.853 + 1.47i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (-5.98 + 10.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (9.03 - 5.21i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.45 + 11.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.26 + 1.30i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.08T + 89T^{2} \)
97 \( 1 + (7.86 - 4.54i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.75275679918112885566721983200, −9.880171969014662771052129679118, −9.358239104763005696017803718801, −8.043834791464976726794878200215, −6.98670501473676425762592311482, −5.90964475421834629959448430147, −5.06451428486831939595929159368, −4.08417094186600899782462148493, −3.18216249982070030647422965579, −1.96267846400148449768514361237, 1.18890334980570827469723251518, 2.27573423656609727293086473417, 4.13707295057665357866898540368, 4.82427388021717488305043112559, 5.86592522189027868109139224053, 6.85636940576017941272786570393, 7.78438159025795481125811942368, 8.116633510296219601842201060739, 9.441777478245181797338261805449, 10.34483674144440142968841600718

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