Properties

Label 2-3360-840.773-c0-0-0
Degree $2$
Conductor $3360$
Sign $-0.772 - 0.635i$
Analytic cond. $1.67685$
Root an. cond. $1.29493$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.258 + 0.965i)3-s + (−0.707 − 0.707i)5-s + (0.866 + 0.5i)7-s + (−0.866 + 0.499i)9-s + (−0.965 + 1.67i)11-s + (0.500 − 0.866i)15-s + (−0.258 + 0.965i)21-s + 1.00i·25-s + (−0.707 − 0.707i)27-s + 0.517i·29-s + (−0.866 − 0.5i)31-s + (−1.86 − 0.500i)33-s + (−0.258 − 0.965i)35-s + (0.965 + 0.258i)45-s + (0.499 + 0.866i)49-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)3-s + (−0.707 − 0.707i)5-s + (0.866 + 0.5i)7-s + (−0.866 + 0.499i)9-s + (−0.965 + 1.67i)11-s + (0.500 − 0.866i)15-s + (−0.258 + 0.965i)21-s + 1.00i·25-s + (−0.707 − 0.707i)27-s + 0.517i·29-s + (−0.866 − 0.5i)31-s + (−1.86 − 0.500i)33-s + (−0.258 − 0.965i)35-s + (0.965 + 0.258i)45-s + (0.499 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.772 - 0.635i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (2033, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ -0.772 - 0.635i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8982038976\)
\(L(\frac12)\) \(\approx\) \(0.8982038976\)
\(L(1)\) 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 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
good11 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T^{2} \)
29 \( 1 - 0.517iT - T^{2} \)
31 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.866 - 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.366 - 0.366i)T + iT^{2} \)
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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

−9.098294708655223576783801678892, −8.336462470415157336173521998503, −7.80712169102451728596012081660, −7.15737707772892731653174114857, −5.69908656418861469216120801088, −5.01956705850234686379004978122, −4.61789140480253257575977642772, −3.84515637862489627195311318068, −2.68330665081589269561858035395, −1.76124191434841000765943120228, 0.49696285530254795849518448935, 1.85552736412119374360199749842, 2.98485407401563782897383173469, 3.48501601351933442731974813860, 4.65332136198908728273540297918, 5.68864435433801558630133753895, 6.31055998223499601769795588757, 7.32852204469118780557139505896, 7.65655044144408990972026174915, 8.379135740400703814642522903725

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