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Properties

Label 2-675-45.34-c1-0-0
Degree $2$
Conductor $675$
Sign $-0.949 - 0.312i$
Analytic cond. $5.38990$
Root an. cond. $2.32161$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.409 + 0.236i)2-s + (−0.888 − 1.53i)4-s + (−2.21 − 1.28i)7-s − 1.78i·8-s + (−3.08 + 5.34i)11-s + (−1.84 + 1.06i)13-s + (−0.606 − 1.05i)14-s + (−1.35 + 2.34i)16-s − 3.16i·17-s − 0.356·19-s + (−2.52 + 1.45i)22-s + (−3.64 + 2.10i)23-s − 1.00·26-s + 4.55i·28-s + (−0.843 + 1.46i)29-s + ⋯
L(s)  = 1  + (0.289 + 0.167i)2-s + (−0.444 − 0.769i)4-s + (−0.838 − 0.484i)7-s − 0.631i·8-s + (−0.929 + 1.61i)11-s + (−0.512 + 0.295i)13-s + (−0.162 − 0.280i)14-s + (−0.338 + 0.585i)16-s − 0.768i·17-s − 0.0817·19-s + (−0.539 + 0.311i)22-s + (−0.760 + 0.439i)23-s − 0.197·26-s + 0.860i·28-s + (−0.156 + 0.271i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.949 - 0.312i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.949 - 0.312i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00479255 + 0.0298733i\)
\(L(\frac12)\) \(\approx\) \(0.00479255 + 0.0298733i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.409 - 0.236i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (2.21 + 1.28i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.08 - 5.34i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.84 - 1.06i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.16iT - 17T^{2} \)
19 \( 1 + 0.356T + 19T^{2} \)
23 \( 1 + (3.64 - 2.10i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.843 - 1.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.12 - 7.15i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.63iT - 37T^{2} \)
41 \( 1 + (1.36 + 2.36i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.64 + 3.83i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (9.89 + 5.71i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.43iT - 53T^{2} \)
59 \( 1 + (5.10 + 8.84i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.00549 + 0.00952i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.851 + 0.491i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.43T + 71T^{2} \)
73 \( 1 + 6.61iT - 73T^{2} \)
79 \( 1 + (4.73 - 8.20i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.02 - 5.20i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.26T + 89T^{2} \)
97 \( 1 + (6.24 + 3.60i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.55108428335730699483629672040, −9.834452476644662197282930726889, −9.642988224952652990610110259563, −8.254388800103488044653285479982, −7.04667847540297128918212479316, −6.63814203149318403384772817520, −5.19353941825656102803070388333, −4.73705509414020676937190086216, −3.46497554190206265444233449150, −1.93593463750177351381607978760, 0.01376099674566402442650471231, 2.61022972439375284405419277240, 3.28696040078547225624440344752, 4.41821778638539719258456263196, 5.63367541461677385109456019208, 6.29474673760238430325952324654, 7.83071396488442861152385757245, 8.242402376419917887038128625100, 9.191164955546722103707173950638, 10.10347121397738717934980526735

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