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Properties

Label 2-1445-5.4-c1-0-59
Degree $2$
Conductor $1445$
Sign $0.998 + 0.0578i$
Analytic cond. $11.5383$
Root an. cond. $3.39681$
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  − 0.951i·2-s + 2.47i·3-s + 1.09·4-s + (−2.23 − 0.129i)5-s + 2.35·6-s + 0.533i·7-s − 2.94i·8-s − 3.12·9-s + (−0.123 + 2.12i)10-s + 4.67·11-s + 2.70i·12-s − 3.92i·13-s + 0.507·14-s + (0.320 − 5.52i)15-s − 0.614·16-s + ⋯
L(s)  = 1  − 0.672i·2-s + 1.42i·3-s + 0.547·4-s + (−0.998 − 0.0578i)5-s + 0.961·6-s + 0.201i·7-s − 1.04i·8-s − 1.04·9-s + (−0.0389 + 0.671i)10-s + 1.40·11-s + 0.781i·12-s − 1.08i·13-s + 0.135·14-s + (0.0826 − 1.42i)15-s − 0.153·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0578i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1445\)    =    \(5 \cdot 17^{2}\)
Sign: $0.998 + 0.0578i$
Analytic conductor: \(11.5383\)
Root analytic conductor: \(3.39681\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1445} (579, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1445,\ (\ :1/2),\ 0.998 + 0.0578i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.862030009\)
\(L(\frac12)\) \(\approx\) \(1.862030009\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.23 + 0.129i)T \)
17 \( 1 \)
good2 \( 1 + 0.951iT - 2T^{2} \)
3 \( 1 - 2.47iT - 3T^{2} \)
7 \( 1 - 0.533iT - 7T^{2} \)
11 \( 1 - 4.67T + 11T^{2} \)
13 \( 1 + 3.92iT - 13T^{2} \)
19 \( 1 - 1.00T + 19T^{2} \)
23 \( 1 + 5.73iT - 23T^{2} \)
29 \( 1 - 8.28T + 29T^{2} \)
31 \( 1 + 2.47T + 31T^{2} \)
37 \( 1 - 4.09iT - 37T^{2} \)
41 \( 1 + 0.814T + 41T^{2} \)
43 \( 1 - 6.43iT - 43T^{2} \)
47 \( 1 - 1.70iT - 47T^{2} \)
53 \( 1 + 8.16iT - 53T^{2} \)
59 \( 1 - 6.05T + 59T^{2} \)
61 \( 1 - 1.13T + 61T^{2} \)
67 \( 1 + 7.68iT - 67T^{2} \)
71 \( 1 + 6.94T + 71T^{2} \)
73 \( 1 - 12.0iT - 73T^{2} \)
79 \( 1 + 1.48T + 79T^{2} \)
83 \( 1 - 16.1iT - 83T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 - 0.228iT - 97T^{2} \)
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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

−9.796240859807401649005950987210, −8.867385809101846745977019708177, −8.183451879967679261243555040058, −7.03412159775428421131239956521, −6.24573314049736054417894586662, −4.99144654767177135467211768379, −4.17346826974794861647296994930, −3.50972961189138216777367028049, −2.75592786252868010874063886950, −0.952918860837954349531699598157, 1.11772119840871184574823370377, 2.10492569881992540097985401747, 3.44275834930147721129756800953, 4.49370464225278018765464629901, 5.86100236632660490162778265716, 6.58000507720531990188579783323, 7.22647118638894366990144789564, 7.45208647918136854893882739796, 8.499693694364000366428876420622, 9.073156352980652040485617529254

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