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Properties

Label 2-1275-85.84-c1-0-19
Degree $2$
Conductor $1275$
Sign $0.650 + 0.759i$
Analytic cond. $10.1809$
Root an. cond. $3.19075$
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  − i·2-s − 3-s + 4-s + i·6-s − 4·7-s − 3i·8-s + 9-s + 4i·11-s − 12-s + 2i·13-s + 4i·14-s − 16-s + (4 − i)17-s − i·18-s + 4·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577·3-s + 0.5·4-s + 0.408i·6-s − 1.51·7-s − 1.06i·8-s + 0.333·9-s + 1.20i·11-s − 0.288·12-s + 0.554i·13-s + 1.06i·14-s − 0.250·16-s + (0.970 − 0.242i)17-s − 0.235i·18-s + 0.917·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1275\)    =    \(3 \cdot 5^{2} \cdot 17\)
Sign: $0.650 + 0.759i$
Analytic conductor: \(10.1809\)
Root analytic conductor: \(3.19075\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1275} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1275,\ (\ :1/2),\ 0.650 + 0.759i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.409737291\)
\(L(\frac12)\) \(\approx\) \(1.409737291\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 \)
17 \( 1 + (-4 + i)T \)
good2 \( 1 + iT - 2T^{2} \)
7 \( 1 + 4T + 7T^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + 8iT - 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 8iT - 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 16T + 97T^{2} \)
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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

−9.626820318860676955253041725028, −9.386088437362252075495074878424, −7.59690116380178581010689529032, −7.02740253551514500392835680108, −6.35046950981171294541604373104, −5.42323179478601760960811715972, −4.16893951858566354955405928345, −3.27238535435631751762886086508, −2.30506904182871039880750210074, −0.899664157972691799955202863838, 0.912227621088752742803498282175, 2.88433016776409213100842773246, 3.45681486625946942967903896636, 5.14049516851820470528905617486, 5.84776591607594587985636413968, 6.36279836599149462874235612059, 7.16122724154490459956563611439, 7.957700128672090165162646483262, 8.885705664535840645517521854953, 9.894946523783105116305876291933

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