Properties

Label 2-675-135.38-c1-0-40
Degree $2$
Conductor $675$
Sign $-0.467 - 0.883i$
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  + (−0.153 − 1.75i)2-s + (−1.70 − 0.305i)3-s + (−1.09 + 0.192i)4-s + (−0.274 + 3.04i)6-s + (1.93 + 2.75i)7-s + (−0.406 − 1.51i)8-s + (2.81 + 1.04i)9-s + (−1.11 − 3.07i)11-s + (1.92 + 0.00542i)12-s + (−5.79 − 0.506i)13-s + (4.55 − 3.81i)14-s + (−4.68 + 1.70i)16-s + (−0.298 + 1.11i)17-s + (1.39 − 5.10i)18-s + (−6.40 − 3.69i)19-s + ⋯
L(s)  = 1  + (−0.108 − 1.24i)2-s + (−0.984 − 0.176i)3-s + (−0.546 + 0.0963i)4-s + (−0.112 + 1.24i)6-s + (0.730 + 1.04i)7-s + (−0.143 − 0.536i)8-s + (0.937 + 0.347i)9-s + (−0.337 − 0.926i)11-s + (0.554 + 0.00156i)12-s + (−1.60 − 0.140i)13-s + (1.21 − 1.02i)14-s + (−1.17 + 0.426i)16-s + (−0.0723 + 0.270i)17-s + (0.329 − 1.20i)18-s + (−1.46 − 0.848i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.467 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.467 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.467 - 0.883i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.467 - 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.160815 + 0.267110i\)
\(L(\frac12)\) \(\approx\) \(0.160815 + 0.267110i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.70 + 0.305i)T \)
5 \( 1 \)
good2 \( 1 + (0.153 + 1.75i)T + (-1.96 + 0.347i)T^{2} \)
7 \( 1 + (-1.93 - 2.75i)T + (-2.39 + 6.57i)T^{2} \)
11 \( 1 + (1.11 + 3.07i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (5.79 + 0.506i)T + (12.8 + 2.25i)T^{2} \)
17 \( 1 + (0.298 - 1.11i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (6.40 + 3.69i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.56 + 1.09i)T + (7.86 + 21.6i)T^{2} \)
29 \( 1 + (0.381 + 0.320i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-0.573 - 3.25i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (4.40 + 1.17i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.39 + 1.65i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-3.62 - 1.68i)T + (27.6 + 32.9i)T^{2} \)
47 \( 1 + (-4.06 + 2.84i)T + (16.0 - 44.1i)T^{2} \)
53 \( 1 + (-1.25 + 1.25i)T - 53iT^{2} \)
59 \( 1 + (9.92 + 3.61i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (1.82 - 10.3i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-0.509 + 5.82i)T + (-65.9 - 11.6i)T^{2} \)
71 \( 1 + (6.62 - 3.82i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.89 - 1.84i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.08 + 6.06i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (-6.22 + 0.544i)T + (81.7 - 14.4i)T^{2} \)
89 \( 1 + (0.260 - 0.450i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.99 - 4.27i)T + (-62.3 - 74.3i)T^{2} \)
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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

−10.41084393710537362592697856519, −9.275104159983632562104746237143, −8.420574604779894858489967046076, −7.20270035080722480206036692266, −6.18154389536404133809189755073, −5.21399501349865615625340070917, −4.34581533309216183765694340163, −2.71963549492043957707230642750, −1.89054733183487838003849814918, −0.17962817777553488003756681912, 2.02318136024966873043845894874, 4.33930824906016727057477023468, 4.79319936958363128894506836736, 5.80044744466341978774849392085, 6.81626620071592548491355430788, 7.40696049325354364504831127969, 7.991978455419785216665285546144, 9.388448706068213337003368870135, 10.27222463737337911933959950055, 10.89542742070450856908161402916

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