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Properties

Label 2-1045-5.4-c1-0-27
Degree $2$
Conductor $1045$
Sign $-0.146 - 0.989i$
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  + 1.28i·2-s − 0.496i·3-s + 0.346·4-s + (−0.328 − 2.21i)5-s + 0.638·6-s + 3.51i·7-s + 3.01i·8-s + 2.75·9-s + (2.84 − 0.421i)10-s − 11-s − 0.171i·12-s + 2.55i·13-s − 4.52·14-s + (−1.09 + 0.162i)15-s − 3.18·16-s + 3.95i·17-s + ⋯
L(s)  = 1  + 0.909i·2-s − 0.286i·3-s + 0.173·4-s + (−0.146 − 0.989i)5-s + 0.260·6-s + 1.32i·7-s + 1.06i·8-s + 0.917·9-s + (0.899 − 0.133i)10-s − 0.301·11-s − 0.0496i·12-s + 0.708i·13-s − 1.20·14-s + (−0.283 + 0.0420i)15-s − 0.796·16-s + 0.960i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.146 - 0.989i$
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: \((2,\ 1045,\ (\ :1/2),\ -0.146 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.814529281\)
\(L(\frac12)\) \(\approx\) \(1.814529281\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.328 + 2.21i)T \)
11 \( 1 + T \)
19 \( 1 - T \)
good2 \( 1 - 1.28iT - 2T^{2} \)
3 \( 1 + 0.496iT - 3T^{2} \)
7 \( 1 - 3.51iT - 7T^{2} \)
13 \( 1 - 2.55iT - 13T^{2} \)
17 \( 1 - 3.95iT - 17T^{2} \)
23 \( 1 + 7.91iT - 23T^{2} \)
29 \( 1 - 3.06T + 29T^{2} \)
31 \( 1 - 1.12T + 31T^{2} \)
37 \( 1 - 6.39iT - 37T^{2} \)
41 \( 1 - 1.65T + 41T^{2} \)
43 \( 1 - 8.34iT - 43T^{2} \)
47 \( 1 - 13.0iT - 47T^{2} \)
53 \( 1 + 7.33iT - 53T^{2} \)
59 \( 1 - 1.66T + 59T^{2} \)
61 \( 1 + 3.64T + 61T^{2} \)
67 \( 1 - 10.5iT - 67T^{2} \)
71 \( 1 - 13.9T + 71T^{2} \)
73 \( 1 + 9.76iT - 73T^{2} \)
79 \( 1 - 5.30T + 79T^{2} \)
83 \( 1 + 3.85iT - 83T^{2} \)
89 \( 1 - 8.82T + 89T^{2} \)
97 \( 1 - 1.91iT - 97T^{2} \)
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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

−9.924527138905193260714506626518, −8.995852337636716603590773100165, −8.314475870826360766097923100040, −7.80053148783832787520878746508, −6.56783566626045147472616708557, −6.14055799302533129352085534895, −5.06319116436276837282760267083, −4.39319743269500850408994089893, −2.60166032268930454966729455100, −1.57108898232929357338549861920, 0.858986239669751084782662851614, 2.27588064683053709176146115156, 3.48265832251740564341730810364, 3.85649966920900823947501305724, 5.17441181796762204877836218751, 6.55862353893125677763141988562, 7.37676658483292730323523102786, 7.57723072900660211065006387829, 9.357904239079376931737164735676, 10.12979480204043991801736109754

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