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Properties

Label 2-3360-840.293-c0-0-9
Degree $2$
Conductor $3360$
Sign $0.973 - 0.229i$
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  + (0.923 + 0.382i)3-s + (0.923 + 0.382i)5-s + (0.707 − 0.707i)7-s + (0.707 + 0.707i)9-s + (−0.541 − 0.541i)13-s + (0.707 + 0.707i)15-s − 1.84i·19-s + (0.923 − 0.382i)21-s + (−1 + i)23-s + (0.707 + 0.707i)25-s + (0.382 + 0.923i)27-s + (0.923 − 0.382i)35-s + (−0.292 − 0.707i)39-s + (0.382 + 0.923i)45-s − 1.00i·49-s + ⋯
L(s)  = 1  + (0.923 + 0.382i)3-s + (0.923 + 0.382i)5-s + (0.707 − 0.707i)7-s + (0.707 + 0.707i)9-s + (−0.541 − 0.541i)13-s + (0.707 + 0.707i)15-s − 1.84i·19-s + (0.923 − 0.382i)21-s + (−1 + i)23-s + (0.707 + 0.707i)25-s + (0.382 + 0.923i)27-s + (0.923 − 0.382i)35-s + (−0.292 − 0.707i)39-s + (0.382 + 0.923i)45-s − 1.00i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.973 - 0.229i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (1553, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ 0.973 - 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.191722508\)
\(L(\frac12)\) \(\approx\) \(2.191722508\)
\(L(1)\) 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 \( 1 + (-0.923 - 0.382i)T \)
5 \( 1 + (-0.923 - 0.382i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good11 \( 1 + T^{2} \)
13 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + 1.84iT - T^{2} \)
23 \( 1 + (1 - i)T - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + 1.84T + T^{2} \)
61 \( 1 + 0.765T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−8.883166406361695719738837841275, −8.090588858044885251980701886424, −7.36544104610469560203597445596, −6.85507619398593051100216084780, −5.66438783638597269822277131366, −4.92412449807120061149027990378, −4.19792346534997651625672872917, −3.13208003276914848998773191894, −2.41571812206722988649321887569, −1.45164395205423187010011638461, 1.60027795934095171860718553013, 2.02104356333769841329614070332, 2.98165141913523582906484344811, 4.16771483394908363987549486887, 4.88521445422036579731262649182, 5.96352728828073566109723017889, 6.35042994814027866804031253820, 7.56096087748299206719074873641, 8.069539001429748004715702400538, 8.791995848079769529551641207198

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