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Properties

Label 2-1232-77.54-c1-0-32
Degree $2$
Conductor $1232$
Sign $-0.498 + 0.866i$
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  + (−1.35 + 0.779i)3-s + (0.882 + 0.509i)5-s + (−2.25 − 1.38i)7-s + (−0.284 + 0.492i)9-s + (0.510 + 3.27i)11-s − 0.167·13-s − 1.58·15-s + (−1.47 − 2.55i)17-s + (0.155 − 0.269i)19-s + (4.12 + 0.108i)21-s + (−0.237 + 0.411i)23-s + (−1.98 − 3.43i)25-s − 5.56i·27-s + 1.89i·29-s + (2.20 − 1.27i)31-s + ⋯
L(s)  = 1  + (−0.779 + 0.450i)3-s + (0.394 + 0.227i)5-s + (−0.852 − 0.522i)7-s + (−0.0947 + 0.164i)9-s + (0.153 + 0.988i)11-s − 0.0463·13-s − 0.410·15-s + (−0.358 − 0.620i)17-s + (0.0357 − 0.0619i)19-s + (0.899 + 0.0236i)21-s + (−0.0495 + 0.0858i)23-s + (−0.396 − 0.686i)25-s − 1.07i·27-s + 0.352i·29-s + (0.396 − 0.228i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.498 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $-0.498 + 0.866i$
Analytic conductor: \(9.83756\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :1/2),\ -0.498 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2348881792\)
\(L(\frac12)\) \(\approx\) \(0.2348881792\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.25 + 1.38i)T \)
11 \( 1 + (-0.510 - 3.27i)T \)
good3 \( 1 + (1.35 - 0.779i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.882 - 0.509i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 0.167T + 13T^{2} \)
17 \( 1 + (1.47 + 2.55i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.155 + 0.269i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.237 - 0.411i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.89iT - 29T^{2} \)
31 \( 1 + (-2.20 + 1.27i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.04 - 5.27i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 + 10.1iT - 43T^{2} \)
47 \( 1 + (-3.28 - 1.89i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.21 + 7.29i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.40 - 3.12i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.93 + 10.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.19 + 8.99i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.5T + 71T^{2} \)
73 \( 1 + (4.85 + 8.40i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.06 + 3.50i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.6T + 83T^{2} \)
89 \( 1 + (-5.97 - 3.45i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.0iT - 97T^{2} \)
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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

−9.785758955768547642824054619473, −8.812106044753273399627044974573, −7.63126962417200265047411808162, −6.78022848909648136480775186269, −6.18051527966998136502695633637, −5.11149688488733387881042294045, −4.44264964897771696349806678649, −3.26733807518677908414843593536, −2.02810480060816880740260191774, −0.11144520200897829275769725990, 1.35103900681898077611443307713, 2.82296010846210824200850417817, 3.81938683247079516342934176921, 5.23567596125297882340254044836, 5.99815789271707386893338138857, 6.34897268012300684514059861367, 7.34657624040151611518694710241, 8.580215042677125336811959232813, 9.075835849236465516030516795402, 9.995031101180481105481928853183

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