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Properties

Label 1-1045-1045.433-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.220 + 0.975i$
Analytic cond. $4.85295$
Root an. cond. $4.85295$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.927 − 0.374i)2-s + (0.275 − 0.961i)3-s + (0.719 + 0.694i)4-s + (−0.615 + 0.788i)6-s + (0.406 − 0.913i)7-s + (−0.406 − 0.913i)8-s + (−0.848 − 0.529i)9-s + (0.866 − 0.5i)12-s + (−0.999 − 0.0348i)13-s + (−0.719 + 0.694i)14-s + (0.0348 + 0.999i)16-s + (−0.529 − 0.848i)17-s + (0.587 + 0.809i)18-s + (−0.766 − 0.642i)21-s + (−0.984 + 0.173i)23-s + (−0.990 + 0.139i)24-s + ⋯
L(s)  = 1  + (−0.927 − 0.374i)2-s + (0.275 − 0.961i)3-s + (0.719 + 0.694i)4-s + (−0.615 + 0.788i)6-s + (0.406 − 0.913i)7-s + (−0.406 − 0.913i)8-s + (−0.848 − 0.529i)9-s + (0.866 − 0.5i)12-s + (−0.999 − 0.0348i)13-s + (−0.719 + 0.694i)14-s + (0.0348 + 0.999i)16-s + (−0.529 − 0.848i)17-s + (0.587 + 0.809i)18-s + (−0.766 − 0.642i)21-s + (−0.984 + 0.173i)23-s + (−0.990 + 0.139i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.220 + 0.975i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.220 + 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1098863972 - 0.1375190017i\)
\(L(\frac12)\) \(\approx\) \(-0.1098863972 - 0.1375190017i\)
\(L(1)\) \(\approx\) \(0.4810993351 - 0.3383322076i\)
\(L(1)\) \(\approx\) \(0.4810993351 - 0.3383322076i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.927 - 0.374i)T \)
3 \( 1 + (0.275 - 0.961i)T \)
7 \( 1 + (0.406 - 0.913i)T \)
13 \( 1 + (-0.999 - 0.0348i)T \)
17 \( 1 + (-0.529 - 0.848i)T \)
23 \( 1 + (-0.984 + 0.173i)T \)
29 \( 1 + (-0.241 + 0.970i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (-0.961 - 0.275i)T \)
43 \( 1 + (0.984 + 0.173i)T \)
47 \( 1 + (0.0697 + 0.997i)T \)
53 \( 1 + (0.469 + 0.882i)T \)
59 \( 1 + (-0.997 - 0.0697i)T \)
61 \( 1 + (0.990 + 0.139i)T \)
67 \( 1 + (0.642 + 0.766i)T \)
71 \( 1 + (0.882 + 0.469i)T \)
73 \( 1 + (0.829 + 0.559i)T \)
79 \( 1 + (-0.615 - 0.788i)T \)
83 \( 1 + (-0.207 - 0.978i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (-0.927 - 0.374i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.9489706575653924771447492619, −21.26775777390552421313185533031, −20.40619416707718434841680801222, −19.74011598146770793265540241035, −19.00192556283342126317185366694, −18.15442169732766739445053708352, −17.23896397101643229264454924120, −16.767248157137667625868072683692, −15.723909866752641317527782924132, −15.1822143647958974915523465627, −14.69979036162015074280109176438, −13.756028717821699864484428301388, −12.28573264024307990854582059221, −11.52989692581559266813573777894, −10.730204829055508274949147773955, −9.85142219004310244121552946974, −9.33502446248226559310724510353, −8.354636337339248129206351140123, −7.97136713125324375484256585301, −6.64976874840558482423541477018, −5.67912153823575569945805975450, −5.01699389062380687992445396644, −3.876780264272290512054380979915, −2.48634910016186323125552197544, −1.968603503538489969468433365085, 0.09286928638595019254695460909, 1.283538351187015495431207337563, 2.13162746841080271372605513251, 3.04851933891627832422901246764, 4.12373367021464406308174134950, 5.48192443029212546327858519921, 6.855919737238418350503707769791, 7.216925989830180520228771181943, 7.94644930450679252239221291872, 8.84372353905602381264487141440, 9.63990113385689778978969216969, 10.62972356372233395848406369378, 11.361494072596608960099315193078, 12.21919588454966545257942358913, 12.83226256522572768553395263097, 13.905638818772310588123688938006, 14.46243926112762309081974504992, 15.688732756783070099862632003594, 16.59376988139979138575250309321, 17.42122939045140244806396418285, 17.85034311944665087355725756778, 18.623768134833587631983350834594, 19.50010464419628741650022734273, 20.115449799490027822860861654507, 20.47192749567279672927213609803

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