Properties

Label 2-975-5.4-c1-0-2
Degree $2$
Conductor $975$
Sign $0.447 + 0.894i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
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  + 2.41i·2-s + i·3-s − 3.82·4-s − 2.41·6-s + 2.82i·7-s − 4.41i·8-s − 9-s − 2·11-s − 3.82i·12-s i·13-s − 6.82·14-s + 2.99·16-s + 3.65i·17-s − 2.41i·18-s − 2.82·19-s + ⋯
L(s)  = 1  + 1.70i·2-s + 0.577i·3-s − 1.91·4-s − 0.985·6-s + 1.06i·7-s − 1.56i·8-s − 0.333·9-s − 0.603·11-s − 1.10i·12-s − 0.277i·13-s − 1.82·14-s + 0.749·16-s + 0.886i·17-s − 0.569i·18-s − 0.648·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.543869 - 0.336129i\)
\(L(\frac12)\) \(\approx\) \(0.543869 - 0.336129i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + iT \)
good2 \( 1 - 2.41iT - 2T^{2} \)
7 \( 1 - 2.82iT - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
17 \( 1 - 3.65iT - 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 6.82T + 31T^{2} \)
37 \( 1 + 3.65iT - 37T^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 - 9.65iT - 43T^{2} \)
47 \( 1 - 0.343iT - 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 - 3.65T + 59T^{2} \)
61 \( 1 + 9.31T + 61T^{2} \)
67 \( 1 + 1.17iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 11.6iT - 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + 7.65iT - 83T^{2} \)
89 \( 1 + 9.17T + 89T^{2} \)
97 \( 1 - 7.65iT - 97T^{2} \)
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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

−10.49370504414654693588041742015, −9.444791562545566582866647218748, −8.802602219291349782385387471679, −8.184213187641571372812647508007, −7.38328318365397481590527685071, −6.18050745953012928292312499152, −5.76812870874958251909192061529, −4.91034121126452710860501708403, −3.98509939943295872239769294276, −2.48329095870409955935573450255, 0.29447920915469462073745020250, 1.54404130552665583382355219566, 2.60186256966660128009654903175, 3.64130468942501621101632294348, 4.49224849754064866300322219559, 5.59148833463464964694904636914, 7.01345150121244380997268399794, 7.66304732669546696913337246137, 8.822577884279626949511310629623, 9.548010188425038212327043437520

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