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Properties

Label 2-2675-535.534-c0-0-7
Degree $2$
Conductor $2675$
Sign $-0.894 + 0.447i$
Analytic cond. $1.33499$
Root an. cond. $1.15542$
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 − 11-s + i·12-s − i·13-s + 16-s + 19-s − i·23-s − i·27-s − 2·29-s + i·33-s + i·37-s − 39-s − 41-s + 44-s + ⋯
L(s)  = 1  − i·3-s − 4-s − 11-s + i·12-s − i·13-s + 16-s + 19-s − i·23-s − i·27-s − 2·29-s + i·33-s + i·37-s − 39-s − 41-s + 44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2675\)    =    \(5^{2} \cdot 107\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(1.33499\)
Root analytic conductor: \(1.15542\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2675} (2674, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2675,\ (\ :0),\ -0.894 + 0.447i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6473785811\)
\(L(\frac12)\) \(\approx\) \(0.6473785811\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
107 \( 1 + iT \)
good2 \( 1 + T^{2} \)
3 \( 1 + iT - T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + iT - T^{2} \)
29 \( 1 + 2T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT - T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 2iT - T^{2} \)
53 \( 1 + iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 - 2iT - T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.410484545635315968975019104707, −8.008846473204623902790954326253, −7.36331874764307049271585839505, −6.48833756483378923178983977608, −5.41372272696846119953645450590, −5.06664773441479224128424485183, −3.83431821229711524136147477100, −2.94413982448561029318408129922, −1.73001420645477234961213114935, −0.43933303292923241120017543792, 1.63338806928681703829034679499, 3.19349810053482707106989396430, 3.85759364844889995229122518943, 4.67867769513034496355445334580, 5.23601019688245605298131521864, 5.98389285101048447572537731265, 7.40378445669419238599952527288, 7.76244038014232274171803021556, 8.992045125414382936730893324842, 9.356866402447006106039389789271

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