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Properties

Label 2-670-67.22-c1-0-18
Degree $2$
Conductor $670$
Sign $-0.926 + 0.375i$
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 + (−0.442 − 3.07i)3-s + (0.841 − 0.540i)4-s + (0.415 − 0.909i)5-s + (−1.29 − 2.83i)6-s + (0.952 − 0.279i)7-s + (0.654 − 0.755i)8-s + (−6.40 + 1.88i)9-s + (0.142 − 0.989i)10-s + (0.513 − 1.12i)11-s + (−2.03 − 2.35i)12-s + (−1.87 − 2.16i)13-s + (0.834 − 0.536i)14-s + (−2.98 − 0.876i)15-s + (0.415 − 0.909i)16-s + (1.78 + 1.14i)17-s + ⋯
L(s)  = 1  + (0.678 − 0.199i)2-s + (−0.255 − 1.77i)3-s + (0.420 − 0.270i)4-s + (0.185 − 0.406i)5-s + (−0.527 − 1.15i)6-s + (0.359 − 0.105i)7-s + (0.231 − 0.267i)8-s + (−2.13 + 0.627i)9-s + (0.0450 − 0.313i)10-s + (0.154 − 0.339i)11-s + (−0.588 − 0.678i)12-s + (−0.520 − 0.600i)13-s + (0.223 − 0.143i)14-s + (−0.770 − 0.226i)15-s + (0.103 − 0.227i)16-s + (0.433 + 0.278i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 + 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(670\)    =    \(2 \cdot 5 \cdot 67\)
Sign: $-0.926 + 0.375i$
Analytic conductor: \(5.34997\)
Root analytic conductor: \(2.31300\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{670} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 670,\ (\ :1/2),\ -0.926 + 0.375i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.384676 - 1.97477i\)
\(L(\frac12)\) \(\approx\) \(0.384676 - 1.97477i\)
\(L(\frac{3}{2})\) 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 \( 1 + (8.07 - 1.34i)T \)
good3 \( 1 + (0.442 + 3.07i)T + (-2.87 + 0.845i)T^{2} \)
7 \( 1 + (-0.952 + 0.279i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (-0.513 + 1.12i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (1.87 + 2.16i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (-1.78 - 1.14i)T + (7.06 + 15.4i)T^{2} \)
19 \( 1 + (-0.440 - 0.129i)T + (15.9 + 10.2i)T^{2} \)
23 \( 1 + (-0.346 - 2.41i)T + (-22.0 + 6.47i)T^{2} \)
29 \( 1 - 1.15T + 29T^{2} \)
31 \( 1 + (0.121 - 0.139i)T + (-4.41 - 30.6i)T^{2} \)
37 \( 1 - 6.05T + 37T^{2} \)
41 \( 1 + (7.00 + 4.50i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (-6.91 - 4.44i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 + (-1.16 - 8.07i)T + (-45.0 + 13.2i)T^{2} \)
53 \( 1 + (-4.98 + 3.20i)T + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (0.771 - 0.890i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (4.23 + 9.26i)T + (-39.9 + 46.1i)T^{2} \)
71 \( 1 + (-8.89 + 5.71i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (0.565 + 1.23i)T + (-47.8 + 55.1i)T^{2} \)
79 \( 1 + (-3.86 - 4.46i)T + (-11.2 + 78.1i)T^{2} \)
83 \( 1 + (-5.25 + 11.5i)T + (-54.3 - 62.7i)T^{2} \)
89 \( 1 + (-2.24 + 15.6i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 - 9.56T + 97T^{2} \)
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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

−10.42760848257817759920814490691, −9.131301463498246520475349599694, −7.949055408420504015037176562929, −7.54245096121984849327516213142, −6.39556890599647286490997826246, −5.76384177603603968309836458226, −4.82567173250107740005241751210, −3.18022144926755051849094017742, −1.99982933265776892506688021290, −0.920534454281706595713086130610, 2.51510420891607363903064211127, 3.64380501122735601452478384767, 4.52768296983768986024963804923, 5.18361092674852696288983863030, 6.11720983114283085617581314396, 7.19521274496684757040312389635, 8.452849065077329473242788392403, 9.415065888370195988259334501483, 10.08856151697166222855392514645, 10.84692223735503228020058596241

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