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Properties

Label 2-1875-15.14-c0-0-1
Degree $2$
Conductor $1875$
Sign $1$
Analytic cond. $0.935746$
Root an. cond. $0.967340$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s − 4-s + 0.618i·7-s − 9-s + i·12-s + 1.61i·13-s + 16-s + 1.61·19-s + 0.618·21-s + i·27-s − 0.618i·28-s + 0.618·31-s + 36-s − 1.61i·37-s + 1.61·39-s + ⋯
L(s)  = 1  − i·3-s − 4-s + 0.618i·7-s − 9-s + i·12-s + 1.61i·13-s + 16-s + 1.61·19-s + 0.618·21-s + i·27-s − 0.618i·28-s + 0.618·31-s + 36-s − 1.61i·37-s + 1.61·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.935746\)
Root analytic conductor: \(0.967340\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1875} (1874, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1875,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8628567175\)
\(L(\frac12)\) \(\approx\) \(0.8628567175\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 \)
good2 \( 1 + T^{2} \)
7 \( 1 - 0.618iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 1.61iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - 1.61T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 0.618T + T^{2} \)
37 \( 1 + 1.61iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.61iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 0.618T + T^{2} \)
67 \( 1 - 0.618iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 0.618iT - T^{2} \)
79 \( 1 + 0.618T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 0.618iT - T^{2} \)
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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

−9.201720902482609424312660961572, −8.755613279097115819907950363152, −7.83715529174849242938487550207, −7.16441026098067904666682152868, −6.19421086918991958981764386873, −5.47731925707286504533175410172, −4.60186262719070004045284443848, −3.52016108802284660815630977638, −2.38711306373040656734786170510, −1.18965090219329615260178866988, 0.812433149108671769061590866278, 3.00974504110567628375561249392, 3.57855881658874899825057807706, 4.55027205063799171549847984934, 5.26510671090073373411532992396, 5.83399331700095886711049526616, 7.24382920992635111204001691775, 8.103061237387423609837890334214, 8.639397566603263282193267310595, 9.625311291415356730553754787499

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