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Properties

Label 2-1232-7.2-c1-0-14
Degree $2$
Conductor $1232$
Sign $-0.198 - 0.980i$
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  + (1.20 + 2.09i)3-s + (0.707 − 1.22i)5-s + (1.62 + 2.09i)7-s + (−1.41 + 2.44i)9-s + (0.5 + 0.866i)11-s − 3.82·13-s + 3.41·15-s + (1 + 1.73i)17-s + (−2.70 + 4.68i)19-s + (−2.41 + 5.91i)21-s + (1.29 − 2.23i)23-s + (1.50 + 2.59i)25-s + 0.414·27-s + 29-s + (0.828 + 1.43i)31-s + ⋯
L(s)  = 1  + (0.696 + 1.20i)3-s + (0.316 − 0.547i)5-s + (0.612 + 0.790i)7-s + (−0.471 + 0.816i)9-s + (0.150 + 0.261i)11-s − 1.06·13-s + 0.881·15-s + (0.242 + 0.420i)17-s + (−0.621 + 1.07i)19-s + (−0.526 + 1.29i)21-s + (0.269 − 0.466i)23-s + (0.300 + 0.519i)25-s + 0.0797·27-s + 0.185·29-s + (0.148 + 0.257i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $-0.198 - 0.980i$
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: \((2,\ 1232,\ (\ :1/2),\ -0.198 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.200991704\)
\(L(\frac12)\) \(\approx\) \(2.200991704\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 2.09i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (-1.20 - 2.09i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.707 + 1.22i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 3.82T + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.70 - 4.68i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.29 + 2.23i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (-0.828 - 1.43i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.53 - 4.39i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.24T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (0.414 - 0.717i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.70 + 4.68i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.792 - 1.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.91 + 11.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.03 + 5.25i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.2T + 71T^{2} \)
73 \( 1 + (-0.292 - 0.507i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.03 - 5.25i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.48T + 83T^{2} \)
89 \( 1 + (-7.41 + 12.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−9.830484249318426962274341680177, −9.167547285244367040521583103403, −8.513178899970937667053662248541, −7.87241982893010613817237659717, −6.55092074386906787347873162157, −5.35913816293157517649850798414, −4.83901836703066958258953984551, −3.95674420380050696630317704880, −2.82957817773784741283924099015, −1.75935902826918923463083187250, 0.879355077469824128975951296143, 2.18627785894050671913497548053, 2.85421765224126827791523280846, 4.22185879222684714702754044908, 5.27921847385394017735834954711, 6.56792313328835432616465678396, 7.11880717009089363543081046710, 7.66643670962944060515552590694, 8.507973083184731381086173641484, 9.362928148905450634726849562595

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