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Properties

Label 2-675-45.32-c1-0-4
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: $\chi_{675} (557, \cdot )$
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

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

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