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Properties

Label 2-1875-5.4-c1-0-8
Degree $2$
Conductor $1875$
Sign $1$
Analytic cond. $14.9719$
Root an. cond. $3.86936$
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  − 2.38i·2-s + i·3-s − 3.69·4-s + 2.38·6-s − 3.31i·7-s + 4.05i·8-s − 9-s − 4.36·11-s − 3.69i·12-s + 5.85i·13-s − 7.91·14-s + 2.28·16-s − 0.407i·17-s + 2.38i·18-s + 6.64·19-s + ⋯
L(s)  = 1  − 1.68i·2-s + 0.577i·3-s − 1.84·4-s + 0.974·6-s − 1.25i·7-s + 1.43i·8-s − 0.333·9-s − 1.31·11-s − 1.06i·12-s + 1.62i·13-s − 2.11·14-s + 0.570·16-s − 0.0989i·17-s + 0.562i·18-s + 1.52·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1875\)    =    \(3 \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(14.9719\)
Root analytic conductor: \(3.86936\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1875} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1875,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7525237833\)
\(L(\frac12)\) \(\approx\) \(0.7525237833\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 \)
good2 \( 1 + 2.38iT - 2T^{2} \)
7 \( 1 + 3.31iT - 7T^{2} \)
11 \( 1 + 4.36T + 11T^{2} \)
13 \( 1 - 5.85iT - 13T^{2} \)
17 \( 1 + 0.407iT - 17T^{2} \)
19 \( 1 - 6.64T + 19T^{2} \)
23 \( 1 - 4.61iT - 23T^{2} \)
29 \( 1 + 3.30T + 29T^{2} \)
31 \( 1 + 8.77T + 31T^{2} \)
37 \( 1 + 3.09iT - 37T^{2} \)
41 \( 1 - 2.89T + 41T^{2} \)
43 \( 1 - 1.33iT - 43T^{2} \)
47 \( 1 - 11.0iT - 47T^{2} \)
53 \( 1 - 2.70iT - 53T^{2} \)
59 \( 1 - 5.80T + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 - 0.0418iT - 67T^{2} \)
71 \( 1 + 16.2T + 71T^{2} \)
73 \( 1 - 5.35iT - 73T^{2} \)
79 \( 1 - 3.55T + 79T^{2} \)
83 \( 1 - 4.98iT - 83T^{2} \)
89 \( 1 - 1.94T + 89T^{2} \)
97 \( 1 + 5.99iT - 97T^{2} \)
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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

−9.504305085911150008265514667671, −9.004586391823893565229286977169, −7.68299577146501342548678552239, −7.15009376943304169443475565996, −5.59468020675956427646589509666, −4.76587521843264995646657771359, −3.97131234450003336906309962535, −3.38808457034617285539637844734, −2.29391702380907758215896553784, −1.16823125538585990392583805427, 0.30225140913211278048843645557, 2.33386183728360862532879477985, 3.31782142504163614024201974054, 4.99129492410109579021501937369, 5.59332441810538991100120981594, 5.76998200001039360301548604500, 7.03226910225066990697472557251, 7.58579769333299090815403032264, 8.262141619562422394467172303447, 8.733211414101563315272320922966

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