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Properties

Label 2-668-668.667-c1-0-41
Degree $2$
Conductor $668$
Sign $0.275 + 0.961i$
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  + (−1.40 + 0.197i)2-s − 2.07i·3-s + (1.92 − 0.551i)4-s + (0.408 + 2.90i)6-s + 4.81i·7-s + (−2.58 + 1.15i)8-s − 1.30·9-s − 6.62i·11-s + (−1.14 − 3.99i)12-s + (−0.949 − 6.74i)14-s + (3.39 − 2.12i)16-s + (1.83 − 0.257i)18-s + 1.14i·19-s + 10.0·21-s + (1.30 + 9.27i)22-s + ⋯
L(s)  = 1  + (−0.990 + 0.139i)2-s − 1.19i·3-s + (0.961 − 0.275i)4-s + (0.166 + 1.18i)6-s + 1.82i·7-s + (−0.913 + 0.407i)8-s − 0.436·9-s − 1.99i·11-s + (−0.330 − 1.15i)12-s + (−0.253 − 1.80i)14-s + (0.847 − 0.530i)16-s + (0.431 − 0.0607i)18-s + 0.263i·19-s + 2.18·21-s + (0.278 + 1.97i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(668\)    =    \(2^{2} \cdot 167\)
Sign: $0.275 + 0.961i$
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: \((2,\ 668,\ (\ :1/2),\ 0.275 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.762557 - 0.574437i\)
\(L(\frac12)\) \(\approx\) \(0.762557 - 0.574437i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.197i)T \)
167 \( 1 + 12.9iT \)
good3 \( 1 + 2.07iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 - 4.81iT - 7T^{2} \)
11 \( 1 + 6.62iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 1.14iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 10.7T + 29T^{2} \)
31 \( 1 + 8.16iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 12.6iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 5.24T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 0.303T + 89T^{2} \)
97 \( 1 + 8.71T + 97T^{2} \)
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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

−10.27763304911133327791813887830, −9.111401343611847873906344296909, −8.428739526134466286886709557279, −8.091475780612222343149237905557, −6.73839682705032431253580385002, −6.12904128579731729059071157888, −5.44351098360715654177734842081, −3.03780270335232468784269990772, −2.22543104464070988115488029082, −0.808367255621283299122456870737, 1.29521671929626548443700499064, 3.04499069809458947044185456654, 4.27155119366777503611073148151, 4.79953317058762833600315122554, 6.80013489470964330150862822369, 7.11333865215318444617039853511, 8.179900115200716651732548618851, 9.329662774613853781401866217597, 9.956174135935917772634725713524, 10.46741878526628550068944213693

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