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Properties

Label 2-3360-105.83-c0-0-7
Degree $2$
Conductor $3360$
Sign $-0.525 + 0.850i$
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 + (−0.707 − 0.707i)7-s − 1.00i·9-s − 1.00i·15-s + (−1.41 + 1.41i)17-s − 1.41·19-s − 1.00·21-s + (1 − i)23-s − 1.00i·25-s + (−0.707 − 0.707i)27-s − 1.41i·31-s − 1.00·35-s + (1 − i)37-s + 1.41·41-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (0.707 − 0.707i)5-s + (−0.707 − 0.707i)7-s − 1.00i·9-s − 1.00i·15-s + (−1.41 + 1.41i)17-s − 1.41·19-s − 1.00·21-s + (1 − i)23-s − 1.00i·25-s + (−0.707 − 0.707i)27-s − 1.41i·31-s − 1.00·35-s + (1 − i)37-s + 1.41·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.525 + 0.850i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (3233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ -0.525 + 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.475945180\)
\(L(\frac12)\) \(\approx\) \(1.475945180\)
\(L(1)\) 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 \( 1 + (0.707 + 0.707i)T \)
good11 \( 1 - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
19 \( 1 + 1.41T + T^{2} \)
23 \( 1 + (-1 + i)T - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 1.41iT - T^{2} \)
37 \( 1 + (-1 + i)T - iT^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
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

−8.608964089648895358324878786181, −7.939208453326992675755090436258, −6.95052263867316841114402604751, −6.35295122920641661242240613270, −5.87555554137069728187084209125, −4.29412392941798015255737200479, −4.10377789202258324898164171755, −2.62247524080756529876846945993, −2.03515831636750882701375497583, −0.75864946296297369517950204106, 1.99740591748223141957097562317, 2.74538573981288335019967272345, 3.26446227484495128333845277901, 4.47406885138366541483869650523, 5.15444193027023143104803891148, 6.12922659319171642054425028512, 6.77285201365310256102065864059, 7.51261322385210389200016227239, 8.647188616933480927540422401622, 9.152762337271154083849193637719

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