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Properties

Label 2-1045-5.4-c1-0-1
Degree $2$
Conductor $1045$
Sign $0.905 + 0.423i$
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  + 2.05i·2-s + 2.80i·3-s − 2.22·4-s + (−2.02 − 0.947i)5-s − 5.76·6-s + 1.02i·7-s − 0.461i·8-s − 4.86·9-s + (1.94 − 4.16i)10-s − 11-s − 6.23i·12-s − 1.06i·13-s − 2.11·14-s + (2.65 − 5.67i)15-s − 3.50·16-s − 1.41i·17-s + ⋯
L(s)  = 1  + 1.45i·2-s + 1.61i·3-s − 1.11·4-s + (−0.905 − 0.423i)5-s − 2.35·6-s + 0.388i·7-s − 0.163i·8-s − 1.62·9-s + (0.616 − 1.31i)10-s − 0.301·11-s − 1.80i·12-s − 0.294i·13-s − 0.565·14-s + (0.686 − 1.46i)15-s − 0.875·16-s − 0.344i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.423i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 + 0.423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.905 + 0.423i$
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: \((2,\ 1045,\ (\ :1/2),\ 0.905 + 0.423i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3761623052\)
\(L(\frac12)\) \(\approx\) \(0.3761623052\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.02 + 0.947i)T \)
11 \( 1 + T \)
19 \( 1 + T \)
good2 \( 1 - 2.05iT - 2T^{2} \)
3 \( 1 - 2.80iT - 3T^{2} \)
7 \( 1 - 1.02iT - 7T^{2} \)
13 \( 1 + 1.06iT - 13T^{2} \)
17 \( 1 + 1.41iT - 17T^{2} \)
23 \( 1 + 3.94iT - 23T^{2} \)
29 \( 1 - 1.85T + 29T^{2} \)
31 \( 1 + 7.16T + 31T^{2} \)
37 \( 1 - 1.81iT - 37T^{2} \)
41 \( 1 - 1.21T + 41T^{2} \)
43 \( 1 - 7.57iT - 43T^{2} \)
47 \( 1 - 9.29iT - 47T^{2} \)
53 \( 1 + 9.70iT - 53T^{2} \)
59 \( 1 + 7.10T + 59T^{2} \)
61 \( 1 + 2.23T + 61T^{2} \)
67 \( 1 + 4.13iT - 67T^{2} \)
71 \( 1 + 15.3T + 71T^{2} \)
73 \( 1 + 6.22iT - 73T^{2} \)
79 \( 1 - 2.22T + 79T^{2} \)
83 \( 1 - 10.1iT - 83T^{2} \)
89 \( 1 - 4.69T + 89T^{2} \)
97 \( 1 - 6.64iT - 97T^{2} \)
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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

−10.64045223665448483777671855833, −9.483243581970564672415520148668, −8.930841691477732228394917992592, −8.218247614114220699052807001858, −7.51451703749997457946681482968, −6.35424358644018467701997475920, −5.38941903649839205996907701165, −4.79903086081559011603723125617, −4.09704032597736699062207011477, −2.91720658988713877538190051343, 0.17238632152101059935510347807, 1.42312398857202301979128059956, 2.39132440757289506601135319625, 3.37962114617681198438896670371, 4.27157261693057377748151903293, 5.79802241937087563890265943843, 7.01766256386390069826221776932, 7.31099857603699241390290986412, 8.311752405078067228292818213383, 9.158160653727760405285423442948

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