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Properties

Label 2-1232-7.2-c1-0-36
Degree $2$
Conductor $1232$
Sign $-0.386 + 0.922i$
Analytic cond. $9.83756$
Root an. cond. $3.13649$
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  + (−0.0990 − 0.171i)3-s + (1.96 − 3.41i)5-s + (−0.167 − 2.64i)7-s + (1.48 − 2.56i)9-s + (0.5 + 0.866i)11-s + 0.692·13-s − 0.780·15-s + (−0.425 − 0.736i)17-s + (0.376 − 0.652i)19-s + (−0.436 + 0.290i)21-s + (−0.321 + 0.556i)23-s + (−5.25 − 9.10i)25-s − 1.18·27-s + 5.43·29-s + (3.22 + 5.58i)31-s + ⋯
L(s)  = 1  + (−0.0571 − 0.0990i)3-s + (0.880 − 1.52i)5-s + (−0.0633 − 0.997i)7-s + (0.493 − 0.854i)9-s + (0.150 + 0.261i)11-s + 0.191·13-s − 0.201·15-s + (−0.103 − 0.178i)17-s + (0.0863 − 0.149i)19-s + (−0.0952 + 0.0633i)21-s + (−0.0670 + 0.116i)23-s + (−1.05 − 1.82i)25-s − 0.227·27-s + 1.00·29-s + (0.578 + 1.00i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(9.83756\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :1/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.902581253\)
\(L(\frac12)\) \(\approx\) \(1.902581253\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (0.167 + 2.64i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (0.0990 + 0.171i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.96 + 3.41i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 0.692T + 13T^{2} \)
17 \( 1 + (0.425 + 0.736i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.376 + 0.652i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.321 - 0.556i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.43T + 29T^{2} \)
31 \( 1 + (-3.22 - 5.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.04 - 8.74i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.89T + 41T^{2} \)
43 \( 1 + 2.97T + 43T^{2} \)
47 \( 1 + (6.33 - 10.9i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.01 - 1.75i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.86 + 6.69i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.74 + 8.21i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.40 + 7.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.313T + 71T^{2} \)
73 \( 1 + (-8.00 - 13.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.77 + 6.54i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.1T + 83T^{2} \)
89 \( 1 + (0.530 - 0.918i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.85T + 97T^{2} \)
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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

−9.639438295861694078447780854914, −8.713979910507979851018393489350, −7.978762374448169401071937828791, −6.77606386300650719698077768159, −6.26225526642006813365061757198, −4.96483364546031924879308255395, −4.53090578533709941475528276917, −3.34128823508960586785468011977, −1.60936787587088560422452538481, −0.861906003136127119256232684271, 1.95710828961918059767454793431, 2.60548399166133743172373616288, 3.69289426852122449675945534058, 5.07638070786186324655293127448, 5.91347809019303504423080457472, 6.53814386467943856373676814874, 7.39616408097397409052339087089, 8.363682252230782988236199410593, 9.298076300389146957733465805726, 10.15210734425019625497396940013

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