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Properties

Label 2-3360-840.293-c0-0-3
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  + (−0.923 + 0.382i)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 − i·15-s − 0.765i·19-s + (0.382 − 0.923i)21-s + (1 − i)23-s + (−0.707 − 0.707i)25-s + (−0.382 + 0.923i)27-s + (−0.382 − 0.923i)35-s + (1.70 + 0.707i)39-s + (0.382 + 0.923i)45-s − 1.00i·49-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)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 − i·15-s − 0.765i·19-s + (0.382 − 0.923i)21-s + (1 − i)23-s + (−0.707 − 0.707i)25-s + (−0.382 + 0.923i)27-s + (−0.382 − 0.923i)35-s + (1.70 + 0.707i)39-s + (0.382 + 0.923i)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: $\chi_{3360} (1553, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ 0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5217809761\)
\(L(\frac12)\) \(\approx\) \(0.5217809761\)
\(L(1)\) 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 \( 1 + (0.923 - 0.382i)T \)
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

−8.838073353409543104981287376248, −7.86100885711562542914360694185, −6.93312311874782241439005281964, −6.65812395965823788660405176834, −5.60782853870331109566493749940, −5.11441074311235905104273159083, −4.11205462903971628560263862571, −3.05530646386958373743540368208, −2.52621335711375944197659963931, −0.42614791313030668874294794406, 1.00991624650945446467931092421, 2.07483691843020813095914769882, 3.58508307847642596690946181672, 4.39339357819245294997815385609, 5.02650582056583415426351446829, 5.80487158921769292527802981935, 6.80597010404950438154875051574, 7.22270600699145042286391603407, 7.906833404315944964374404582791, 8.951788769498723536009461388451

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