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Properties

Label 2-668-668.667-c1-0-79
Degree $2$
Conductor $668$
Sign $-0.990 - 0.136i$
Analytic cond. $5.33400$
Root an. cond. $2.30954$
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  + (1.06 − 0.929i)2-s − 2.77i·3-s + (0.272 − 1.98i)4-s + (−2.57 − 2.95i)6-s − 0.0155i·7-s + (−1.55 − 2.36i)8-s − 4.68·9-s − 0.605i·11-s + (−5.49 − 0.756i)12-s + (−0.0144 − 0.0165i)14-s + (−3.85 − 1.08i)16-s + (−4.99 + 4.35i)18-s + 3.70i·19-s − 0.0430·21-s + (−0.562 − 0.644i)22-s + ⋯
L(s)  = 1  + (0.753 − 0.657i)2-s − 1.60i·3-s + (0.136 − 0.990i)4-s + (−1.05 − 1.20i)6-s − 0.00586i·7-s + (−0.548 − 0.836i)8-s − 1.56·9-s − 0.182i·11-s + (−1.58 − 0.218i)12-s + (−0.00385 − 0.00442i)14-s + (−0.962 − 0.270i)16-s + (−1.17 + 1.02i)18-s + 0.849i·19-s − 0.00939·21-s + (−0.119 − 0.137i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(668\)    =    \(2^{2} \cdot 167\)
Sign: $-0.990 - 0.136i$
Analytic conductor: \(5.33400\)
Root analytic conductor: \(2.30954\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{668} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 668,\ (\ :1/2),\ -0.990 - 0.136i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.138569 + 2.02313i\)
\(L(\frac12)\) \(\approx\) \(0.138569 + 2.02313i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.06 + 0.929i)T \)
167 \( 1 - 12.9iT \)
good3 \( 1 + 2.77iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + 0.0155iT - 7T^{2} \)
11 \( 1 + 0.605iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 3.70iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 4.56T + 29T^{2} \)
31 \( 1 + 11.0iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 7.81iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 9.17T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 5.02T + 89T^{2} \)
97 \( 1 - 18.7T + 97T^{2} \)
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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

−10.33209598017812872808875190501, −9.232453531990841559909797336494, −8.146712680410122566577097772058, −7.26833215431647649270279685215, −6.32425453442349362109470740690, −5.74142837353055721575654126693, −4.41094646673404363461234873891, −3.04723679444509397407902554520, −2.04387321913718930113189690914, −0.874172771618075267497780396086, 2.80745462233013809628590616894, 3.68309135133952221406302065353, 4.75852188112699712263012669572, 5.11899856910112533135597419629, 6.35881502700970459812010262207, 7.29325654007801028141792192672, 8.637011108458194506793100234912, 9.006324872446239464501378285377, 10.21934852194953962169021129086, 10.84086160682880540510554031382

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