↑

Properties

Label 2-1150-115.22-c1-0-21
Degree $2$
Conductor $1150$
Sign $0.708 + 0.706i$
Analytic cond. $9.18279$
Root an. cond. $3.03031$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + (1.22 − 1.22i)3-s + 1.00i·4-s − 1.73·6-s + (1.50 − 1.50i)7-s + (0.707 − 0.707i)8-s + 5.03i·11-s + (1.22 + 1.22i)12-s + (−3.34 + 3.34i)13-s − 2.12·14-s − 1.00·16-s + (5.06 − 5.06i)17-s + 5.03·19-s − 3.68i·21-s + (3.56 − 3.56i)22-s + (3.80 + 2.91i)23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.707 − 0.707i)3-s + 0.500i·4-s − 0.707·6-s + (0.569 − 0.569i)7-s + (0.250 − 0.250i)8-s + 1.51i·11-s + (0.353 + 0.353i)12-s + (−0.928 + 0.928i)13-s − 0.569·14-s − 0.250·16-s + (1.22 − 1.22i)17-s + 1.15·19-s − 0.804i·21-s + (0.759 − 0.759i)22-s + (0.794 + 0.607i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 + 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.708 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $0.708 + 0.706i$
Analytic conductor: \(9.18279\)
Root analytic conductor: \(3.03031\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1150} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1150,\ (\ :1/2),\ 0.708 + 0.706i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.767753595\)
\(L(\frac12)\) \(\approx\) \(1.767753595\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
23 \( 1 + (-3.80 - 2.91i)T \)
good3 \( 1 + (-1.22 + 1.22i)T - 3iT^{2} \)
7 \( 1 + (-1.50 + 1.50i)T - 7iT^{2} \)
11 \( 1 - 5.03iT - 11T^{2} \)
13 \( 1 + (3.34 - 3.34i)T - 13iT^{2} \)
17 \( 1 + (-5.06 + 5.06i)T - 17iT^{2} \)
19 \( 1 - 5.03T + 19T^{2} \)
29 \( 1 + 5.46iT - 29T^{2} \)
31 \( 1 - 4.73T + 31T^{2} \)
37 \( 1 + (-4.11 + 4.11i)T - 37iT^{2} \)
41 \( 1 + 11.9T + 41T^{2} \)
43 \( 1 + (1.10 + 1.10i)T + 43iT^{2} \)
47 \( 1 + (3.34 + 3.34i)T + 47iT^{2} \)
53 \( 1 + (-9.73 - 9.73i)T + 53iT^{2} \)
59 \( 1 + 0.535iT - 59T^{2} \)
61 \( 1 + 3.68iT - 61T^{2} \)
67 \( 1 + (2.05 - 2.05i)T - 67iT^{2} \)
71 \( 1 - 1.07T + 71T^{2} \)
73 \( 1 + (-0.568 + 0.568i)T - 73iT^{2} \)
79 \( 1 - 2.70T + 79T^{2} \)
83 \( 1 + (11.3 + 11.3i)T + 83iT^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 + (-1.10 + 1.10i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.772322648440518690668791887803, −8.982488708994489355556867631522, −7.82607837954032922233854371887, −7.38238094555867317241187298855, −7.02424175595216554463162428451, −5.16019756905450950330247129267, −4.45876390958007905437596035652, −3.06984751324514861118445204938, −2.15186161392460824341045600335, −1.21465450102686913086961458047, 1.06604561344271333853877089250, 2.86569030841930079833956018643, 3.49999681627905408041835875843, 5.00313008709541436144797067269, 5.55577315810128863002670399748, 6.58439514412455923829250568282, 7.81231177817166426113499805145, 8.420215307852136102586547577862, 8.806874764246741994230252447597, 9.987800790578319060151781552471

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