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Properties

Label 2-3360-840.797-c0-0-7
Degree $2$
Conductor $3360$
Sign $0.850 - 0.525i$
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 + (0.923 − 0.382i)5-s + (0.707 + 0.707i)7-s + (−0.707 − 0.707i)9-s + (0.541 − 0.541i)13-s + i·15-s − 1.84i·19-s + (−0.923 + 0.382i)21-s + (1 + i)23-s + (0.707 − 0.707i)25-s + (0.923 − 0.382i)27-s + (0.923 + 0.382i)35-s + (0.292 + 0.707i)39-s + (−0.923 − 0.382i)45-s + 1.00i·49-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)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 + i·15-s − 1.84i·19-s + (−0.923 + 0.382i)21-s + (1 + i)23-s + (0.707 − 0.707i)25-s + (0.923 − 0.382i)27-s + (0.923 + 0.382i)35-s + (0.292 + 0.707i)39-s + (−0.923 − 0.382i)45-s + 1.00i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.850 - 0.525i$
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: \((2,\ 3360,\ (\ :0),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.468426475\)
\(L(\frac12)\) \(\approx\) \(1.468426475\)
\(L(1)\) 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 \( 1 + (-0.923 + 0.382i)T \)
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

−8.972760941171005662411024696185, −8.496199480940910817844939054723, −7.35648921225188337581976519863, −6.34389843049648464660473066124, −5.67150064270666747825106507370, −5.05228368168595529806479811941, −4.57357641575370660172279261210, −3.27661768792086453020602870728, −2.47456290526138598080963817604, −1.15725372841162236500924153982, 1.26734127037171324145746222791, 1.87043369842926759192626975472, 2.97682029829373304520609476845, 4.13674529492336538905655925669, 5.12044433418423793017800443700, 5.85724019512523959164707851334, 6.54093352734078475857173819529, 7.10620575359232123259996256638, 7.955564392918601077381147217543, 8.526798526295033375284647327168

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