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Properties

Label 2-3456-3.2-c0-0-5
Degree $2$
Conductor $3456$
Sign $i$
Analytic cond. $1.72476$
Root an. cond. $1.31330$
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  + 1.41i·5-s − 7-s − 1.41i·11-s − 13-s − 1.41i·17-s − 19-s − 1.41i·23-s − 1.00·25-s − 1.41i·35-s + 37-s − 1.41i·47-s + 2.00·55-s + 1.41i·59-s − 61-s − 1.41i·65-s + ⋯
L(s)  = 1  + 1.41i·5-s − 7-s − 1.41i·11-s − 13-s − 1.41i·17-s − 19-s − 1.41i·23-s − 1.00·25-s − 1.41i·35-s + 37-s − 1.41i·47-s + 2.00·55-s + 1.41i·59-s − 61-s − 1.41i·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3456\)    =    \(2^{7} \cdot 3^{3}\)
Sign: $i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3456} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3456,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5831478671\)
\(L(\frac12)\) \(\approx\) \(0.5831478671\)
\(L(1)\) 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 \( 1 \)
good5 \( 1 - 1.41iT - T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 + T + T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 1.41iT - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−8.646134655813458677582068632251, −7.69657782120303514330329589374, −6.93406280562101360463988895279, −6.47440679853551414278508577157, −5.82955146176077241217108390991, −4.73267427731579002687020529550, −3.68305019835648303843662384563, −2.81391335198223566334849011186, −2.53240608110851098504445717612, −0.33069781107050089668547974991, 1.44508157658753034565196483862, 2.34426137512757194724728187637, 3.66589125921340452212942718224, 4.43934498104254149264951226480, 5.02595662797311093499427230639, 5.96638227411586285224282553694, 6.66804076912655303561602008968, 7.62649973444068756024373245908, 8.152420386851775454725424654418, 9.177907826164004383664034718660

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