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Properties

Label 2-1232-7.4-c1-0-33
Degree $2$
Conductor $1232$
Sign $-0.991 + 0.126i$
Analytic cond. $9.83756$
Root an. cond. $3.13649$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (−1 − 1.73i)5-s + (−2.5 − 0.866i)7-s + (1 + 1.73i)9-s + (−0.5 + 0.866i)11-s + 3·13-s + 1.99·15-s + (1 − 1.73i)17-s + (2 + 3.46i)19-s + (2 − 1.73i)21-s + (−2 − 3.46i)23-s + (0.500 − 0.866i)25-s − 5·27-s − 7·29-s + (−4 + 6.92i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.447 − 0.774i)5-s + (−0.944 − 0.327i)7-s + (0.333 + 0.577i)9-s + (−0.150 + 0.261i)11-s + 0.832·13-s + 0.516·15-s + (0.242 − 0.420i)17-s + (0.458 + 0.794i)19-s + (0.436 − 0.377i)21-s + (−0.417 − 0.722i)23-s + (0.100 − 0.173i)25-s − 0.962·27-s − 1.29·29-s + (−0.718 + 1.24i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(9.83756\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1232,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7T + 29T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6 + 10.3i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (5 + 8.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7 - 12.1i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (2 - 3.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.5 + 14.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−9.329933630163556129072047486157, −8.630049228452528259932130671459, −7.67582800010611480841370463373, −6.92176266594279234629306708142, −5.77518303391107513240947367317, −5.04124582972576391114636920272, −4.06858744375042879372624427463, −3.37374701955847795608371540461, −1.66272708077020189325072904002, 0, 1.68425286801187229954033408362, 3.37465427735161400154629537640, 3.52928926040052736987646460800, 5.22223467949035846085669114292, 6.23717570792415287986264649643, 6.67883005791140474158658675120, 7.49465592859873800125967126719, 8.390736581366374744341806203498, 9.475239784673377497441863918672

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