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Properties

Label 2-65-65.33-c1-0-0
Degree $2$
Conductor $65$
Sign $-0.943 + 0.331i$
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  + (−1.32 + 2.29i)2-s + (−1.25 + 0.335i)3-s + (−2.51 − 4.34i)4-s + (−1.30 + 1.81i)5-s + (0.889 − 3.31i)6-s + (0.0972 − 0.0561i)7-s + 8.00·8-s + (−1.14 + 0.658i)9-s + (−2.44 − 5.39i)10-s + (0.479 + 1.78i)11-s + (4.60 + 4.60i)12-s + (−2.05 + 2.96i)13-s + 0.297i·14-s + (1.02 − 2.71i)15-s + (−5.58 + 9.67i)16-s + (−0.706 + 2.63i)17-s + ⋯
L(s)  = 1  + (−0.936 + 1.62i)2-s + (−0.723 + 0.193i)3-s + (−1.25 − 2.17i)4-s + (−0.583 + 0.812i)5-s + (0.363 − 1.35i)6-s + (0.0367 − 0.0212i)7-s + 2.83·8-s + (−0.380 + 0.219i)9-s + (−0.771 − 1.70i)10-s + (0.144 + 0.539i)11-s + (1.32 + 1.32i)12-s + (−0.569 + 0.821i)13-s + 0.0795i·14-s + (0.264 − 0.700i)15-s + (−1.39 + 2.41i)16-s + (−0.171 + 0.639i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.943 + 0.331i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ -0.943 + 0.331i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0542445 - 0.318213i\)
\(L(\frac12)\) \(\approx\) \(0.0542445 - 0.318213i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.30 - 1.81i)T \)
13 \( 1 + (2.05 - 2.96i)T \)
good2 \( 1 + (1.32 - 2.29i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.25 - 0.335i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-0.0972 + 0.0561i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.479 - 1.78i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.706 - 2.63i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-6.72 - 1.80i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.831 + 3.10i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-4.03 - 2.32i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.624 + 0.624i)T + 31iT^{2} \)
37 \( 1 + (-1.27 - 0.737i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.24 - 1.40i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.76 - 1.00i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 0.345iT - 47T^{2} \)
53 \( 1 + (3.59 + 3.59i)T + 53iT^{2} \)
59 \( 1 + (0.332 - 1.24i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.39 - 2.41i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.0721 + 0.124i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.41 - 5.28i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 9.06T + 73T^{2} \)
79 \( 1 + 15.1iT - 79T^{2} \)
83 \( 1 - 8.53iT - 83T^{2} \)
89 \( 1 + (0.549 - 0.147i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-7.48 - 12.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−15.80883072612207049442452810325, −14.67676933293233094171053581351, −14.08113173349803630023808123916, −11.90494108686658222309032214429, −10.66329742545710858286515630350, −9.660747909882002666376390816408, −8.207412440807301793973566359795, −7.14889594022003820896972724201, −6.17366154594001505570276340646, −4.76687575926219517105024435460, 0.69015128767995660615855814687, 3.25438657632327129915650462578, 5.11592994255804372372170940551, 7.63313443536284094110244653241, 8.783124263337482551814262598044, 9.805514671828349615847562509059, 11.22372888794148685590734507932, 11.78193395837327717988515284329, 12.54427105664559255731963420677, 13.70961244097819901024793336515

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