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Properties

Label 2-3360-840.293-c0-0-11
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: $\chi_{3360} (1553, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ -0.973 + 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9489801452\)
\(L(\frac12)\) \(\approx\) \(0.9489801452\)
\(L(1)\) 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

−8.220725803110875997509599055887, −7.70704559180253351261511859472, −7.34361108865518630303659211422, −6.37422552127631811990744904537, −5.33667137630832365099942128043, −4.49038104802090756636012496738, −3.68317865678735530973936744716, −2.86572896875207406676855758742, −1.69352324724406189620353224899, −0.48976174427291262581174966561, 2.16157231065177360550707366089, 2.84176195310126380926635618530, 3.46433824453624529971639297433, 4.54888044872619501393177881343, 5.19414552198861979278733770581, 6.07003763132055325354077506711, 7.10010125038075689756101032043, 7.920550857778597481748143644525, 8.288460466444111777525642644058, 9.057773567035041290430131479257

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