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Properties

Label 2-667-29.28-c1-0-51
Degree $2$
Conductor $667$
Sign $0.716 - 0.697i$
Analytic cond. $5.32602$
Root an. cond. $2.30781$
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  − 2.34i·2-s − 1.29i·3-s − 3.51·4-s − 0.970·5-s − 3.04·6-s − 2.50·7-s + 3.55i·8-s + 1.31·9-s + 2.27i·10-s + 0.324i·11-s + 4.56i·12-s − 5.03·13-s + 5.87i·14-s + 1.26i·15-s + 1.32·16-s + 1.68i·17-s + ⋯
L(s)  = 1  − 1.66i·2-s − 0.749i·3-s − 1.75·4-s − 0.433·5-s − 1.24·6-s − 0.945·7-s + 1.25i·8-s + 0.437·9-s + 0.720i·10-s + 0.0979i·11-s + 1.31i·12-s − 1.39·13-s + 1.56i·14-s + 0.325i·15-s + 0.330·16-s + 0.408i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $0.716 - 0.697i$
Analytic conductor: \(5.32602\)
Root analytic conductor: \(2.30781\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 667,\ (\ :1/2),\ 0.716 - 0.697i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.192301 + 0.0782036i\)
\(L(\frac12)\) \(\approx\) \(0.192301 + 0.0782036i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 - T \)
29 \( 1 + (-3.85 + 3.75i)T \)
good2 \( 1 + 2.34iT - 2T^{2} \)
3 \( 1 + 1.29iT - 3T^{2} \)
5 \( 1 + 0.970T + 5T^{2} \)
7 \( 1 + 2.50T + 7T^{2} \)
11 \( 1 - 0.324iT - 11T^{2} \)
13 \( 1 + 5.03T + 13T^{2} \)
17 \( 1 - 1.68iT - 17T^{2} \)
19 \( 1 - 4.05iT - 19T^{2} \)
31 \( 1 - 1.70iT - 31T^{2} \)
37 \( 1 + 1.09iT - 37T^{2} \)
41 \( 1 + 1.99iT - 41T^{2} \)
43 \( 1 - 2.73iT - 43T^{2} \)
47 \( 1 + 5.86iT - 47T^{2} \)
53 \( 1 + 1.99T + 53T^{2} \)
59 \( 1 + 13.6T + 59T^{2} \)
61 \( 1 + 2.02iT - 61T^{2} \)
67 \( 1 + 5.17T + 67T^{2} \)
71 \( 1 + 5.91T + 71T^{2} \)
73 \( 1 + 2.80iT - 73T^{2} \)
79 \( 1 + 11.0iT - 79T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 - 9.78iT - 89T^{2} \)
97 \( 1 - 6.46iT - 97T^{2} \)
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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

−9.971580119442306046744081417294, −9.336838121646401108071796091192, −8.079042901307751682566010193549, −7.23172289308851523391457909170, −6.19900959847650379293666540189, −4.68253729484987698959073135538, −3.76181754824389344781077883322, −2.71418484472474981773269821718, −1.65337204084669363181274030181, −0.10919766766044814969609192920, 3.01637556491501115474299770124, 4.37074448470254463313808956843, 4.91413282743525908510033375896, 6.02281895797962875488823769836, 7.05490931751898869148633686521, 7.42628606908044853697122140432, 8.604240272289432204623245277936, 9.501538686514049931753156367866, 9.887934775055332352469713349194, 11.10826581322888239212183641791

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