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Properties

Label 2-296-37.36-c1-0-5
Degree $2$
Conductor $296$
Sign $0.945 - 0.324i$
Analytic cond. $2.36357$
Root an. cond. $1.53739$
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  + 1.93·3-s + 1.58i·5-s + 3.41·7-s + 0.729·9-s − 2.90·11-s − 0.613i·13-s + 3.06i·15-s − 2.09i·17-s + 6.72i·19-s + 6.58·21-s − 6.51i·23-s + 2.47·25-s − 4.38·27-s − 7.97i·29-s + 5.35i·31-s + ⋯
L(s)  = 1  + 1.11·3-s + 0.710i·5-s + 1.28·7-s + 0.243·9-s − 0.876·11-s − 0.170i·13-s + 0.791i·15-s − 0.507i·17-s + 1.54i·19-s + 1.43·21-s − 1.35i·23-s + 0.495·25-s − 0.843·27-s − 1.48i·29-s + 0.962i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 - 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $0.945 - 0.324i$
Analytic conductor: \(2.36357\)
Root analytic conductor: \(1.53739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{296} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 296,\ (\ :1/2),\ 0.945 - 0.324i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85657 + 0.309675i\)
\(L(\frac12)\) \(\approx\) \(1.85657 + 0.309675i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 + (1.97 + 5.75i)T \)
good3 \( 1 - 1.93T + 3T^{2} \)
5 \( 1 - 1.58iT - 5T^{2} \)
7 \( 1 - 3.41T + 7T^{2} \)
11 \( 1 + 2.90T + 11T^{2} \)
13 \( 1 + 0.613iT - 13T^{2} \)
17 \( 1 + 2.09iT - 17T^{2} \)
19 \( 1 - 6.72iT - 19T^{2} \)
23 \( 1 + 6.51iT - 23T^{2} \)
29 \( 1 + 7.97iT - 29T^{2} \)
31 \( 1 - 5.35iT - 31T^{2} \)
41 \( 1 + 0.755T + 41T^{2} \)
43 \( 1 - 0.229iT - 43T^{2} \)
47 \( 1 + 3.95T + 47T^{2} \)
53 \( 1 + 2.23T + 53T^{2} \)
59 \( 1 - 10.0iT - 59T^{2} \)
61 \( 1 + 3.89iT - 61T^{2} \)
67 \( 1 + 3.89T + 67T^{2} \)
71 \( 1 - 2.94T + 71T^{2} \)
73 \( 1 + 8.08T + 73T^{2} \)
79 \( 1 + 1.03iT - 79T^{2} \)
83 \( 1 + 17.0T + 83T^{2} \)
89 \( 1 + 8.80iT - 89T^{2} \)
97 \( 1 - 3.45iT - 97T^{2} \)
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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

−11.73440777077177379448289671789, −10.75268627347561952570839289213, −10.03943609940285427092517838800, −8.670065134536922244999564868212, −8.079468114572666896856347161717, −7.31412064559313375831605264263, −5.81132865894193925594106938566, −4.52126739851937292145703771364, −3.12186639027770530718832129651, −2.09764002429421684948792833945, 1.71235094011254318496823660981, 3.07724444741790325675617359648, 4.59814935992063566343153686191, 5.39561936553429542504530039067, 7.20974887065391294115681163674, 8.144882607714899551131374154214, 8.675759068979466972584660547367, 9.549654531418198618059841106692, 10.88825567363426761457869318989, 11.60998015942984896577789640062

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