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Properties

Label 2-672-21.17-c1-0-3
Degree $2$
Conductor $672$
Sign $-0.832 - 0.553i$
Analytic cond. $5.36594$
Root an. cond. $2.31645$
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 + 1.5i)3-s + (0.5 + 0.866i)5-s + (−1.73 + 2i)7-s + (−1.5 + 2.59i)9-s + (−4.33 − 2.5i)11-s + 3.46i·13-s + (−0.866 + 1.5i)15-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (−4.5 − 0.866i)21-s + (−4.33 + 2.5i)23-s + (2 − 3.46i)25-s − 5.19·27-s + 3.46i·29-s + (7.79 + 4.5i)31-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)3-s + (0.223 + 0.387i)5-s + (−0.654 + 0.755i)7-s + (−0.5 + 0.866i)9-s + (−1.30 − 0.753i)11-s + 0.960i·13-s + (−0.223 + 0.387i)15-s + (0.121 − 0.210i)17-s + (−0.198 + 0.114i)19-s + (−0.981 − 0.188i)21-s + (−0.902 + 0.521i)23-s + (0.400 − 0.692i)25-s − 1.00·27-s + 0.643i·29-s + (1.39 + 0.808i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.832 - 0.553i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.832 - 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.351889 + 1.16437i\)
\(L(\frac12)\) \(\approx\) \(0.351889 + 1.16437i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.866 - 1.5i)T \)
7 \( 1 + (1.73 - 2i)T \)
good5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.33 + 2.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.33 - 2.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.46iT - 29T^{2} \)
31 \( 1 + (-7.79 - 4.5i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (0.866 + 1.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.5 - 6.06i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.33 - 7.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (10.5 - 6.06i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.06 - 10.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2iT - 71T^{2} \)
73 \( 1 + (-7.5 - 4.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.866 + 1.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + (6.5 + 11.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.46iT - 97T^{2} \)
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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

−10.52368310518545071563519193364, −10.12481366840234126274742621048, −9.074492034166327897501117085303, −8.537363813630642116331959370105, −7.48245719645510557073892918207, −6.20917660725994702488636681610, −5.46075418081510686421819452661, −4.33049066356088925436563635132, −3.08476327213726297135428719840, −2.41907790935511280199417700708, 0.57546424641085421603792033567, 2.21894742463176567994385859965, 3.24538114315672058162579459365, 4.57803508742264549528698043728, 5.80757646793603827737710735017, 6.67089949470635481937626698001, 7.79467628025344833621764385252, 8.027703445320769983322539971903, 9.361127368844070729859705380630, 10.07056305566583197723800024353

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