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Properties

Label 2-675-45.38-c1-0-8
Degree $2$
Conductor $675$
Sign $0.961 - 0.275i$
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  + (−2.24 + 0.601i)2-s + (2.93 − 1.69i)4-s + (0.201 + 0.751i)7-s + (−2.29 + 2.29i)8-s + (0.220 + 0.127i)11-s + (0.992 − 3.70i)13-s + (−0.903 − 1.56i)14-s + (0.367 − 0.636i)16-s + (3.93 + 3.93i)17-s − 0.440i·19-s + (−0.570 − 0.152i)22-s + (−3.42 − 0.917i)23-s + 8.90i·26-s + (1.86 + 1.86i)28-s + (2.76 − 4.78i)29-s + ⋯
L(s)  = 1  + (−1.58 + 0.425i)2-s + (1.46 − 0.848i)4-s + (0.0761 + 0.284i)7-s + (−0.809 + 0.809i)8-s + (0.0663 + 0.0383i)11-s + (0.275 − 1.02i)13-s + (−0.241 − 0.418i)14-s + (0.0918 − 0.159i)16-s + (0.953 + 0.953i)17-s − 0.101i·19-s + (−0.121 − 0.0325i)22-s + (−0.713 − 0.191i)23-s + 1.74i·26-s + (0.352 + 0.352i)28-s + (0.513 − 0.888i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.961 - 0.275i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.961 - 0.275i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.692938 + 0.0972409i\)
\(L(\frac12)\) \(\approx\) \(0.692938 + 0.0972409i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.24 - 0.601i)T + (1.73 - i)T^{2} \)
7 \( 1 + (-0.201 - 0.751i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.220 - 0.127i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.992 + 3.70i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-3.93 - 3.93i)T + 17iT^{2} \)
19 \( 1 + 0.440iT - 19T^{2} \)
23 \( 1 + (3.42 + 0.917i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-2.76 + 4.78i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.0971 + 0.168i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.123 + 0.123i)T - 37iT^{2} \)
41 \( 1 + (-3.88 + 2.24i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.33 - 0.357i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-4.17 + 1.11i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.938 + 0.938i)T - 53iT^{2} \)
59 \( 1 + (-4.02 - 6.96i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.44 - 2.49i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.9 - 3.47i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 2.15iT - 71T^{2} \)
73 \( 1 + (-9.18 - 9.18i)T + 73iT^{2} \)
79 \( 1 + (-11.9 - 6.91i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.39 + 5.20i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 0.285T + 89T^{2} \)
97 \( 1 + (-2.34 - 8.73i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.24885569959283592245291086833, −9.712766081384260335739654486713, −8.601743497918575354875006426822, −8.144410307194478518502657912760, −7.35794129048607965808418076785, −6.25958481849516106858877186417, −5.53809027471192438771506781874, −3.87560921621319488627437109405, −2.31851453421113678293979837986, −0.878638174171326884852117245950, 0.976264113792388984190713344813, 2.21876848834731776544357818628, 3.54739817982586546913077579481, 4.95481716486134301568510145651, 6.38483521124006656216967541977, 7.30305284356937684387267023284, 7.983633950010252539926134889390, 8.941931685118170936959715015621, 9.547348460580448485653676219371, 10.29642754055973813808050836699

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