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Properties

Label 2-65-65.58-c1-0-0
Degree $2$
Conductor $65$
Sign $0.243 - 0.969i$
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  + (−1.94 − 1.12i)2-s + (−1.91 + 0.514i)3-s + (1.53 + 2.65i)4-s + (0.247 + 2.22i)5-s + (4.31 + 1.15i)6-s + (0.638 + 1.10i)7-s − 2.39i·8-s + (0.820 − 0.473i)9-s + (2.01 − 4.61i)10-s + (−5.27 + 1.41i)11-s + (−4.30 − 4.30i)12-s + (0.840 + 3.50i)13-s − 2.87i·14-s + (−1.61 − 4.13i)15-s + (0.365 − 0.633i)16-s + (0.833 − 3.11i)17-s + ⋯
L(s)  = 1  + (−1.37 − 0.795i)2-s + (−1.10 + 0.296i)3-s + (0.766 + 1.32i)4-s + (0.110 + 0.993i)5-s + (1.76 + 0.472i)6-s + (0.241 + 0.418i)7-s − 0.848i·8-s + (0.273 − 0.157i)9-s + (0.638 − 1.45i)10-s + (−1.59 + 0.426i)11-s + (−1.24 − 1.24i)12-s + (0.233 + 0.972i)13-s − 0.768i·14-s + (−0.417 − 1.06i)15-s + (0.0913 − 0.158i)16-s + (0.202 − 0.754i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.243 - 0.969i$
Analytic conductor: \(0.519027\)
Root analytic conductor: \(0.720435\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :1/2),\ 0.243 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.209861 + 0.163726i\)
\(L(\frac12)\) \(\approx\) \(0.209861 + 0.163726i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.247 - 2.22i)T \)
13 \( 1 + (-0.840 - 3.50i)T \)
good2 \( 1 + (1.94 + 1.12i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (1.91 - 0.514i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-0.638 - 1.10i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.27 - 1.41i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.833 + 3.11i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.315 - 1.17i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.0428 - 0.160i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-8.41 - 4.85i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.233 + 0.233i)T - 31iT^{2} \)
37 \( 1 + (0.660 - 1.14i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.129 - 0.483i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (6.43 + 1.72i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 3.20T + 47T^{2} \)
53 \( 1 + (-4.49 - 4.49i)T + 53iT^{2} \)
59 \( 1 + (-0.00222 - 0.000595i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.695 + 1.20i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.26 - 3.03i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-11.7 - 3.14i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 7.34iT - 73T^{2} \)
79 \( 1 - 11.1iT - 79T^{2} \)
83 \( 1 - 2.65T + 83T^{2} \)
89 \( 1 + (1.86 + 6.96i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (3.62 - 2.09i)T + (48.5 - 84.0i)T^{2} \)
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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

−15.58372623576431775175201470835, −14.02835760319184292999596481876, −12.19773843375063432507288365063, −11.37694251325229826918616489587, −10.57959876051602881273262504468, −9.887387222981667421713863819041, −8.338742449027689767474033238370, −6.95535542419160601216952249749, −5.26470360573775992349521606933, −2.55562805364076167316249836118, 0.65019814425119579221552571730, 5.22164505054384313860014890614, 6.21107807032089977202002818942, 7.80507747797278923864986551280, 8.501783995512282496609027833806, 10.13117630822186338082392612616, 10.86524162715263386110926899575, 12.39344664004860195814981279248, 13.39916220171541092563569844437, 15.36547872382118152591712055459

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