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Properties

Label 2-675-45.4-c1-0-1
Degree $2$
Conductor $675$
Sign $-0.720 - 0.693i$
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  + (0.495 − 0.285i)2-s + (−0.836 + 1.44i)4-s + (−1.23 + 0.714i)7-s + 2.10i·8-s + (1.33 + 2.31i)11-s + (−4.04 − 2.33i)13-s + (−0.408 + 0.707i)14-s + (−1.07 − 1.85i)16-s + 2.67i·17-s − 4.67·19-s + (1.32 + 0.764i)22-s + (−5.12 − 2.95i)23-s − 2.67·26-s − 2.38i·28-s + (4.74 + 8.21i)29-s + ⋯
L(s)  = 1  + (0.350 − 0.202i)2-s + (−0.418 + 0.724i)4-s + (−0.467 + 0.269i)7-s + 0.742i·8-s + (0.402 + 0.697i)11-s + (−1.12 − 0.648i)13-s + (−0.109 + 0.189i)14-s + (−0.267 − 0.464i)16-s + 0.648i·17-s − 1.07·19-s + (0.282 + 0.162i)22-s + (−1.06 − 0.616i)23-s − 0.524·26-s − 0.451i·28-s + (0.881 + 1.52i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.720 - 0.693i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.720 - 0.693i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.307914 + 0.763443i\)
\(L(\frac12)\) \(\approx\) \(0.307914 + 0.763443i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.495 + 0.285i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (1.23 - 0.714i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.33 - 2.31i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.04 + 2.33i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.67iT - 17T^{2} \)
19 \( 1 + 4.67T + 19T^{2} \)
23 \( 1 + (5.12 + 2.95i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.74 - 8.21i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.48 - 6.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.81iT - 37T^{2} \)
41 \( 1 + (0.735 - 1.27i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.408 + 0.235i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.02 + 3.47i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.14iT - 53T^{2} \)
59 \( 1 + (-0.571 + 0.990i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.26 - 2.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.70 + 3.29i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 + 1.71iT - 73T^{2} \)
79 \( 1 + (0.143 + 0.249i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.71 + 2.14i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (6.78 - 3.91i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.74675060192496133105956804303, −10.03082613084276140855232772394, −9.005960634362272166183443946929, −8.319218078781684938995358255898, −7.34255801927787355493603113980, −6.41075267928912474213610017737, −5.15197504062840538721228612075, −4.33435394269826559582992432426, −3.27034700068217218713044027006, −2.18762100714440226755490809950, 0.37260945716558856067113823118, 2.21492195938791763814519823464, 3.83943252957921974206013136116, 4.56792169991693596368170468883, 5.74839188310735677681533154292, 6.42405380607036286980753449243, 7.35938524965085973691116127494, 8.535785920929532355372574370522, 9.595103506966064763864888507412, 9.885185902641750760706644999392

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