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Properties

Label 2-65-65.28-c1-0-0
Degree $2$
Conductor $65$
Sign $-0.920 - 0.389i$
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  + (−1.58 + 0.915i)2-s + (−0.512 + 1.91i)3-s + (0.677 − 1.17i)4-s + (−1.69 − 1.45i)5-s + (−0.939 − 3.50i)6-s + (−1.76 + 3.06i)7-s − 1.18i·8-s + (−0.803 − 0.463i)9-s + (4.02 + 0.752i)10-s + (−1.00 + 3.74i)11-s + (1.89 + 1.89i)12-s + (3.55 + 0.573i)13-s − 6.48i·14-s + (3.65 − 2.50i)15-s + (2.43 + 4.22i)16-s + (1.95 − 0.524i)17-s + ⋯
L(s)  = 1  + (−1.12 + 0.647i)2-s + (−0.296 + 1.10i)3-s + (0.338 − 0.586i)4-s + (−0.759 − 0.650i)5-s + (−0.383 − 1.43i)6-s + (−0.668 + 1.15i)7-s − 0.417i·8-s + (−0.267 − 0.154i)9-s + (1.27 + 0.237i)10-s + (−0.302 + 1.12i)11-s + (0.548 + 0.548i)12-s + (0.987 + 0.158i)13-s − 1.73i·14-s + (0.943 − 0.646i)15-s + (0.609 + 1.05i)16-s + (0.474 − 0.127i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 - 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.920 - 0.389i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ -0.920 - 0.389i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0742943 + 0.366334i\)
\(L(\frac12)\) \(\approx\) \(0.0742943 + 0.366334i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.69 + 1.45i)T \)
13 \( 1 + (-3.55 - 0.573i)T \)
good2 \( 1 + (1.58 - 0.915i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.512 - 1.91i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.76 - 3.06i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.00 - 3.74i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.95 + 0.524i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.518 + 0.139i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.294 - 0.0788i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (1.71 - 0.988i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.13 + 4.13i)T - 31iT^{2} \)
37 \( 1 + (2.70 + 4.69i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.649 - 0.174i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.28 + 8.51i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 - 9.75T + 47T^{2} \)
53 \( 1 + (-3.16 - 3.16i)T + 53iT^{2} \)
59 \( 1 + (-3.14 - 11.7i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.44 + 2.49i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.98 - 1.14i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.19 + 4.46i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 - 14.7iT - 73T^{2} \)
79 \( 1 + 1.59iT - 79T^{2} \)
83 \( 1 + 7.57T + 83T^{2} \)
89 \( 1 + (-4.54 - 1.21i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-15.4 - 8.91i)T + (48.5 + 84.0i)T^{2} \)
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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

−15.58889035890790376269563038411, −15.27316515842380731306948013031, −12.95203252724114795905826976518, −11.93705215965510210579679536694, −10.41473005650955471054159426602, −9.399065341526052250432511064529, −8.706243224987868848761683826985, −7.33844025933667672669975168555, −5.60567978189764065776146279166, −4.00586583625019691104856269451, 0.825299262928373689453297878164, 3.35407750516282253100758337501, 6.30335023263000964958806013194, 7.48541467670205242362675799339, 8.396533198741097357053555217979, 10.13581975746233514763501029364, 10.94948930386634429625156237549, 11.84580238297364720325552578486, 13.19870946700500525050698984194, 14.10394852528711739880069772234

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