Properties

Label 2-65-65.29-c1-0-0
Degree $2$
Conductor $65$
Sign $-0.606 - 0.795i$
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  + (−0.286 + 0.165i)2-s + (−2.33 + 1.34i)3-s + (−0.945 + 1.63i)4-s + (−2.12 − 0.702i)5-s + (0.445 − 0.771i)6-s + (2.90 + 1.67i)7-s − 1.28i·8-s + (2.12 − 3.67i)9-s + (0.724 − 0.149i)10-s + (1.62 + 2.81i)11-s − 5.08i·12-s + (−1.21 + 3.39i)13-s − 1.10·14-s + (5.89 − 1.21i)15-s + (−1.67 − 2.90i)16-s + (−1.68 − 0.974i)17-s + ⋯
L(s)  = 1  + (−0.202 + 0.116i)2-s + (−1.34 + 0.777i)3-s + (−0.472 + 0.818i)4-s + (−0.949 − 0.314i)5-s + (0.181 − 0.314i)6-s + (1.09 + 0.633i)7-s − 0.455i·8-s + (0.707 − 1.22i)9-s + (0.229 − 0.0474i)10-s + (0.489 + 0.847i)11-s − 1.46i·12-s + (−0.337 + 0.941i)13-s − 0.296·14-s + (1.52 − 0.314i)15-s + (−0.419 − 0.726i)16-s + (−0.409 − 0.236i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.606 - 0.795i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ -0.606 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.192394 + 0.388796i\)
\(L(\frac12)\) \(\approx\) \(0.192394 + 0.388796i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.12 + 0.702i)T \)
13 \( 1 + (1.21 - 3.39i)T \)
good2 \( 1 + (0.286 - 0.165i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (2.33 - 1.34i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-2.90 - 1.67i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.62 - 2.81i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.68 + 0.974i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.622 - 1.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.33 + 1.34i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.78T + 31T^{2} \)
37 \( 1 + (1.68 - 0.974i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.39 + 2.40i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.56 - 4.36i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.86iT - 47T^{2} \)
53 \( 1 + 12.8iT - 53T^{2} \)
59 \( 1 + (1.26 - 2.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.74 + 6.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.47 + 2.00i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.62 - 4.54i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.46iT - 73T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 - 8.61iT - 83T^{2} \)
89 \( 1 + (-5.15 - 8.93i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.56 + 2.63i)T + (48.5 + 84.0i)T^{2} \)
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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

−15.61703367585328640435035925215, −14.49718988483571849131302767755, −12.51116230643452266441542191191, −11.86303998792983587957546828916, −11.15612630686160274648146170458, −9.496413295712411115377695744181, −8.377548243774131858252818496901, −6.95198613835808197715235223469, −4.90208496999044460051823758533, −4.30109500823462424454792714020, 0.833996455731397483900533023978, 4.57110447163098693915333919774, 5.84617707723322058004690151773, 7.22185038281702213599623430821, 8.448165733602766473545901379670, 10.54649057985499888585304347241, 11.10073357244982102106072193939, 11.94884489238834689329668313801, 13.36042420901991252190247549014, 14.48565935464465634040492473057

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