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Properties

Label 2-3360-840.293-c0-0-1
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  + (−0.923 − 0.382i)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 + (0.707 + 0.707i)15-s + 1.84i·19-s + (−0.923 + 0.382i)21-s + (−1 + i)23-s + (0.707 + 0.707i)25-s + (−0.382 − 0.923i)27-s + (−0.923 + 0.382i)35-s + (−0.292 − 0.707i)39-s + (−0.382 − 0.923i)45-s − 1.00i·49-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)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 + (0.707 + 0.707i)15-s + 1.84i·19-s + (−0.923 + 0.382i)21-s + (−1 + i)23-s + (0.707 + 0.707i)25-s + (−0.382 − 0.923i)27-s + (−0.923 + 0.382i)35-s + (−0.292 − 0.707i)39-s + (−0.382 − 0.923i)45-s − 1.00i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

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

\(L(\frac{1}{2})\) \(\approx\) \(0.7822643368\)
\(L(\frac12)\) \(\approx\) \(0.7822643368\)
\(L(1)\) 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 \( 1 + (0.923 + 0.382i)T \)
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

−8.493468660835387657422145276203, −7.977860098679575395274735440686, −7.42087823347574052468484019021, −6.65597085047542635699006685518, −5.73849125966987744657956432597, −5.08804196021603773504475164680, −4.02696128023889474325705469580, −3.81511949704210866873632560155, −1.89008385081340470696412097661, −1.07697561750994030661509749579, 0.66058366343829626933164759105, 2.28981581832743426789709247341, 3.34844838304635508756046542305, 4.32518644132577334412914234709, 4.87441371073937185683998871691, 5.71146910231483742461236121147, 6.51588057685523025689980920026, 7.17355426028736612146142782560, 8.106887480379466983372439208106, 8.667031603173324102604297460613

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