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Properties

Label 2-675-15.8-c1-0-15
Degree $2$
Conductor $675$
Sign $0.973 + 0.229i$
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  + (0.880 + 0.880i)2-s − 0.449i·4-s + (0.224 − 0.224i)7-s + (2.15 − 2.15i)8-s − 3.91i·11-s + (3.22 + 3.22i)13-s + 0.395·14-s + 2.89·16-s + (−3.91 − 3.91i)17-s − 3i·19-s + (3.44 − 3.44i)22-s + (−0.880 + 0.880i)23-s + 5.67i·26-s + (−0.101 − 0.101i)28-s + 7.43·29-s + ⋯
L(s)  = 1  + (0.622 + 0.622i)2-s − 0.224i·4-s + (0.0849 − 0.0849i)7-s + (0.762 − 0.762i)8-s − 1.18i·11-s + (0.894 + 0.894i)13-s + 0.105·14-s + 0.724·16-s + (−0.950 − 0.950i)17-s − 0.688i·19-s + (0.735 − 0.735i)22-s + (−0.183 + 0.183i)23-s + 1.11i·26-s + (−0.0190 − 0.0190i)28-s + 1.38·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.973 + 0.229i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.973 + 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16071 - 0.251580i\)
\(L(\frac12)\) \(\approx\) \(2.16071 - 0.251580i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.880 - 0.880i)T + 2iT^{2} \)
7 \( 1 + (-0.224 + 0.224i)T - 7iT^{2} \)
11 \( 1 + 3.91iT - 11T^{2} \)
13 \( 1 + (-3.22 - 3.22i)T + 13iT^{2} \)
17 \( 1 + (3.91 + 3.91i)T + 17iT^{2} \)
19 \( 1 + 3iT - 19T^{2} \)
23 \( 1 + (0.880 - 0.880i)T - 23iT^{2} \)
29 \( 1 - 7.43T + 29T^{2} \)
31 \( 1 - 9.34T + 31T^{2} \)
37 \( 1 + (1.67 - 1.67i)T - 37iT^{2} \)
41 \( 1 - 1.36iT - 41T^{2} \)
43 \( 1 + (-1.44 - 1.44i)T + 43iT^{2} \)
47 \( 1 + (-3.52 - 3.52i)T + 47iT^{2} \)
53 \( 1 + (5.19 - 5.19i)T - 53iT^{2} \)
59 \( 1 + 13.9T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + (-3.77 + 3.77i)T - 67iT^{2} \)
71 \( 1 + 9.20iT - 71T^{2} \)
73 \( 1 + (4.67 + 4.67i)T + 73iT^{2} \)
79 \( 1 - 5.44iT - 79T^{2} \)
83 \( 1 + (3.03 - 3.03i)T - 83iT^{2} \)
89 \( 1 + 5.28T + 89T^{2} \)
97 \( 1 + (10.1 - 10.1i)T - 97iT^{2} \)
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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

−10.66566860182660299546334807012, −9.474729914335100460370706423382, −8.739313204465546694574556432717, −7.69375292881585524796474518886, −6.44737338651084697725408270128, −6.29233118187871196565803621293, −4.92582234969648022661978665684, −4.27593671686820238937476843026, −2.89802433601626935590418911500, −1.08104442099818857518593651876, 1.72131462526024312967685766431, 2.86427287380545648201944976475, 4.00796006719401609515718863369, 4.71336510370785031716389389007, 5.91165041910262977827383677749, 6.96542035672283289266206386009, 8.137023210319191336331968806271, 8.557636113026930341919664392446, 10.06019012026724289524079156233, 10.56142848123908471402987006528

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