Properties

Label 2-1875-5.4-c1-0-68
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  − 1.33i·2-s i·3-s + 0.209·4-s − 1.33·6-s − 1.27i·7-s − 2.95i·8-s − 9-s + 1.16·11-s − 0.209i·12-s − 3.61i·13-s − 1.71·14-s − 3.53·16-s − 5.35i·17-s + 1.33i·18-s + 7.61·19-s + ⋯
L(s)  = 1  − 0.946i·2-s − 0.577i·3-s + 0.104·4-s − 0.546·6-s − 0.483i·7-s − 1.04i·8-s − 0.333·9-s + 0.351·11-s − 0.0603i·12-s − 1.00i·13-s − 0.457·14-s − 0.884·16-s − 1.29i·17-s + 0.315i·18-s + 1.74·19-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

\(L(1)\) \(\approx\) \(1.924760873\)
\(L(\frac12)\) \(\approx\) \(1.924760873\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.33iT - 2T^{2} \)
7 \( 1 + 1.27iT - 7T^{2} \)
11 \( 1 - 1.16T + 11T^{2} \)
13 \( 1 + 3.61iT - 13T^{2} \)
17 \( 1 + 5.35iT - 17T^{2} \)
19 \( 1 - 7.61T + 19T^{2} \)
23 \( 1 - 3.41iT - 23T^{2} \)
29 \( 1 - 3.21T + 29T^{2} \)
31 \( 1 + 1.09T + 31T^{2} \)
37 \( 1 - 7.80iT - 37T^{2} \)
41 \( 1 + 3.00T + 41T^{2} \)
43 \( 1 + 3.42iT - 43T^{2} \)
47 \( 1 - 9.41iT - 47T^{2} \)
53 \( 1 + 7.64iT - 53T^{2} \)
59 \( 1 + 12.7T + 59T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 + 8.78iT - 67T^{2} \)
71 \( 1 + 15.0T + 71T^{2} \)
73 \( 1 - 13.1iT - 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 + 10.7iT - 83T^{2} \)
89 \( 1 - 4.51T + 89T^{2} \)
97 \( 1 + 18.0iT - 97T^{2} \)
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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

−9.111394777679575291877749270014, −7.80919173660905130032768186065, −7.37940920749346082423022414271, −6.59267253363031864876852652335, −5.59753180957973703665001209240, −4.62782041379142123181982688487, −3.25337931791167998180932131530, −2.97342836069414674962874083967, −1.54449945278199051507580817155, −0.72285824220768535732954385637, 1.70938655456432917214830872471, 2.87492663882899426339007512298, 4.00762694214716184648384810664, 4.93300027979495888700153469136, 5.77782973016628037050269823385, 6.35444779989534810617122143430, 7.22604848720868004699621401617, 7.982343107763876992987226300812, 8.878766653460956498953266951924, 9.269649760133107759468665934130

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