Properties

Label 2-3360-840.797-c0-0-4
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  + (−0.707 − 0.707i)3-s + (0.707 − 0.707i)5-s + i·7-s + 1.00i·9-s + 1.41·11-s − 1.00·15-s + (0.707 − 0.707i)21-s − 1.00i·25-s + (0.707 − 0.707i)27-s + 1.41i·29-s + 2i·31-s + (−1.00 − 1.00i)33-s + (0.707 + 0.707i)35-s + (0.707 + 0.707i)45-s − 49-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s + (0.707 − 0.707i)5-s + i·7-s + 1.00i·9-s + 1.41·11-s − 1.00·15-s + (0.707 − 0.707i)21-s − 1.00i·25-s + (0.707 − 0.707i)27-s + 1.41i·29-s + 2i·31-s + (−1.00 − 1.00i)33-s + (0.707 + 0.707i)35-s + (0.707 + 0.707i)45-s − 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: $\chi_{3360} (2897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ 0.973 + 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.232452340\)
\(L(\frac12)\) \(\approx\) \(1.232452340\)
\(L(1)\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 - iT \)
good11 \( 1 - 1.41T + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 - 2iT - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1 + i)T - iT^{2} \)
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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

−8.739693109551092308687939849226, −8.237518684135137311410493135255, −6.88534397447320625573183634673, −6.64680864283811820581289763685, −5.64060467304776256336655797492, −5.27409178840062880015855129755, −4.36764642988661584798371395177, −3.04580474335413517861486799073, −1.87219009593183543242796165403, −1.24973401361455021484305002856, 0.973027253471665838444414465238, 2.30594605071210871582426499340, 3.68271564241593316547300509276, 4.00508583512675145322602768476, 4.99406100000956186268089200398, 6.04811878382530264620116410581, 6.38663482944322760886361868085, 7.12241744041476748099488694228, 7.994566622135988761074239981782, 9.258903019933022436290966426891

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