Properties

Label 2-3360-840.293-c0-0-10
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  + (0.382 − 0.923i)3-s + (0.382 − 0.923i)5-s + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)9-s + (−1.30 − 1.30i)13-s + (−0.707 − 0.707i)15-s − 0.765i·19-s + (0.382 + 0.923i)21-s + (−1 + i)23-s + (−0.707 − 0.707i)25-s + (−0.923 + 0.382i)27-s + (0.382 + 0.923i)35-s + (−1.70 + 0.707i)39-s + (−0.923 + 0.382i)45-s − 1.00i·49-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)3-s + (0.382 − 0.923i)5-s + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)9-s + (−1.30 − 1.30i)13-s + (−0.707 − 0.707i)15-s − 0.765i·19-s + (0.382 + 0.923i)21-s + (−1 + i)23-s + (−0.707 − 0.707i)25-s + (−0.923 + 0.382i)27-s + (0.382 + 0.923i)35-s + (−1.70 + 0.707i)39-s + (−0.923 + 0.382i)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

\(L(\frac{1}{2})\) \(\approx\) \(0.8809963272\)
\(L(\frac12)\) \(\approx\) \(0.8809963272\)
\(L(1)\) 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 \( 1 + (-0.382 + 0.923i)T \)
5 \( 1 + (-0.382 + 0.923i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good11 \( 1 + T^{2} \)
13 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + 0.765iT - T^{2} \)
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 \( 1 + 0.765T + T^{2} \)
61 \( 1 - 1.84T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
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

−8.379954362573332187573700168137, −7.79625855899921417127556118416, −7.06773381439859358103250311105, −6.09299481778311582799073631086, −5.57527701464035414065763250527, −4.82641110205460797884850059785, −3.48060568041013044083468479434, −2.64455850369572046137065960068, −1.88016560052046086808786165243, −0.44975563607529380306180208946, 2.07791087825325673859182953490, 2.76303946059306675947502956107, 3.80189580948706992139046807584, 4.26778430824464556279750274449, 5.29806775247541127164966108816, 6.26021945969318893011993531049, 6.87885949617520292903665412353, 7.60396368388938306965773854533, 8.466659907264580366414160900812, 9.489648348613677903659007160616

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