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Properties

Label 2-1045-5.4-c1-0-82
Degree $2$
Conductor $1045$
Sign $-0.585 + 0.810i$
Analytic cond. $8.34436$
Root an. cond. $2.88866$
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  + 0.712i·2-s − 3.32i·3-s + 1.49·4-s + (1.31 − 1.81i)5-s + 2.36·6-s − 2.70i·7-s + 2.48i·8-s − 8.03·9-s + (1.29 + 0.933i)10-s + 11-s − 4.95i·12-s + 3.93i·13-s + 1.92·14-s + (−6.02 − 4.35i)15-s + 1.21·16-s − 4.53i·17-s + ⋯
L(s)  = 1  + 0.503i·2-s − 1.91i·3-s + 0.746·4-s + (0.585 − 0.810i)5-s + 0.966·6-s − 1.02i·7-s + 0.879i·8-s − 2.67·9-s + (0.408 + 0.295i)10-s + 0.301·11-s − 1.43i·12-s + 1.09i·13-s + 0.514·14-s + (−1.55 − 1.12i)15-s + 0.303·16-s − 1.10i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 + 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.585 + 0.810i$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ -0.585 + 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.030250908\)
\(L(\frac12)\) \(\approx\) \(2.030250908\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.31 + 1.81i)T \)
11 \( 1 - T \)
19 \( 1 - T \)
good2 \( 1 - 0.712iT - 2T^{2} \)
3 \( 1 + 3.32iT - 3T^{2} \)
7 \( 1 + 2.70iT - 7T^{2} \)
13 \( 1 - 3.93iT - 13T^{2} \)
17 \( 1 + 4.53iT - 17T^{2} \)
23 \( 1 + 3.13iT - 23T^{2} \)
29 \( 1 - 7.37T + 29T^{2} \)
31 \( 1 + 6.00T + 31T^{2} \)
37 \( 1 + 7.56iT - 37T^{2} \)
41 \( 1 + 0.108T + 41T^{2} \)
43 \( 1 - 3.35iT - 43T^{2} \)
47 \( 1 - 7.06iT - 47T^{2} \)
53 \( 1 - 12.1iT - 53T^{2} \)
59 \( 1 + 3.83T + 59T^{2} \)
61 \( 1 + 5.74T + 61T^{2} \)
67 \( 1 - 15.8iT - 67T^{2} \)
71 \( 1 + 7.83T + 71T^{2} \)
73 \( 1 + 11.6iT - 73T^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 - 11.0iT - 83T^{2} \)
89 \( 1 - 7.90T + 89T^{2} \)
97 \( 1 + 19.3iT - 97T^{2} \)
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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

−9.274285151086474561225877212775, −8.576054034003802346817172945970, −7.54578861816282462884511031614, −7.19822329061749184515150335033, −6.41510875705007390308166948835, −5.80595521123053479123835524652, −4.61114655598519344565355890582, −2.78508239733746066875865387953, −1.80599012767945582536566431470, −0.906532090200920137242917607659, 2.14608677807287825624116541737, 3.17208333046743062491190321080, 3.57707776993460697520350473987, 5.07653937050471171187548729293, 5.80718138268462476853594895473, 6.47157086460401408535513089260, 7.971235934054851706214472135975, 8.919912678430676260961094803384, 9.697883093806624380410287290933, 10.37932722025262671202268938674

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