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Properties

Label 2-65-5.4-c1-0-5
Degree $2$
Conductor $65$
Sign $-0.662 + 0.749i$
Analytic cond. $0.519027$
Root an. cond. $0.720435$
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  − 2.67i·2-s − 0.481i·3-s − 5.15·4-s + (1.67 + 1.48i)5-s − 1.28·6-s − 0.806i·7-s + 8.44i·8-s + 2.76·9-s + (3.96 − 4.48i)10-s − 3.67·11-s + 2.48i·12-s − i·13-s − 2.15·14-s + (0.712 − 0.806i)15-s + 12.2·16-s + 1.35i·17-s + ⋯
L(s)  = 1  − 1.89i·2-s − 0.277i·3-s − 2.57·4-s + (0.749 + 0.662i)5-s − 0.525·6-s − 0.304i·7-s + 2.98i·8-s + 0.922·9-s + (1.25 − 1.41i)10-s − 1.10·11-s + 0.716i·12-s − 0.277i·13-s − 0.576·14-s + (0.184 − 0.208i)15-s + 3.06·16-s + 0.327i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.662 + 0.749i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ -0.662 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.354218 - 0.786041i\)
\(L(\frac12)\) \(\approx\) \(0.354218 - 0.786041i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.67 - 1.48i)T \)
13 \( 1 + iT \)
good2 \( 1 + 2.67iT - 2T^{2} \)
3 \( 1 + 0.481iT - 3T^{2} \)
7 \( 1 + 0.806iT - 7T^{2} \)
11 \( 1 + 3.67T + 11T^{2} \)
17 \( 1 - 1.35iT - 17T^{2} \)
19 \( 1 - 1.67T + 19T^{2} \)
23 \( 1 - 6.48iT - 23T^{2} \)
29 \( 1 + 2.41T + 29T^{2} \)
31 \( 1 + 5.28T + 31T^{2} \)
37 \( 1 + 3.76iT - 37T^{2} \)
41 \( 1 + 8.31T + 41T^{2} \)
43 \( 1 + 6.79iT - 43T^{2} \)
47 \( 1 + 3.19iT - 47T^{2} \)
53 \( 1 - 5.73iT - 53T^{2} \)
59 \( 1 + 5.98T + 59T^{2} \)
61 \( 1 + 1.76T + 61T^{2} \)
67 \( 1 + 9.89iT - 67T^{2} \)
71 \( 1 - 8.56T + 71T^{2} \)
73 \( 1 + 11.7iT - 73T^{2} \)
79 \( 1 - 2.26T + 79T^{2} \)
83 \( 1 - 3.84iT - 83T^{2} \)
89 \( 1 + 2.77T + 89T^{2} \)
97 \( 1 - 1.87iT - 97T^{2} \)
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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

−13.74481344790317902117727778451, −13.29439790080493060087555043216, −12.28868952321884398013474176993, −10.89762629119522298221460593506, −10.28363942215524618960099815232, −9.339820764192456405679431270287, −7.55140802681601215059430881173, −5.32702882842433263307798493824, −3.50995963329224430598839239799, −1.91909961512465625173433187138, 4.60448092783219497061505181542, 5.51425556819673294298540917976, 6.85064122668876594296469897895, 8.140655141354650636749691191532, 9.227957740993045543645525656790, 10.16229805151599716975830888844, 12.68895429672687877718789726047, 13.37102741820397249382349950608, 14.47206298657666482533181744103, 15.54673816780158929299622790499

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