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Properties

Label 1-1012-1012.735-r0-0-0
Degree $1$
Conductor $1012$
Sign $0.935 + 0.352i$
Analytic cond. $4.69970$
Root an. cond. $4.69970$
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  + (−0.309 + 0.951i)3-s + (0.809 − 0.587i)5-s + (0.309 + 0.951i)7-s + (−0.809 − 0.587i)9-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)15-s + (0.809 − 0.587i)17-s + (0.309 − 0.951i)19-s − 21-s + (0.309 − 0.951i)25-s + (0.809 − 0.587i)27-s + (0.309 + 0.951i)29-s + (0.809 + 0.587i)31-s + (0.809 + 0.587i)35-s + (−0.309 − 0.951i)37-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)3-s + (0.809 − 0.587i)5-s + (0.309 + 0.951i)7-s + (−0.809 − 0.587i)9-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)15-s + (0.809 − 0.587i)17-s + (0.309 − 0.951i)19-s − 21-s + (0.309 − 0.951i)25-s + (0.809 − 0.587i)27-s + (0.309 + 0.951i)29-s + (0.809 + 0.587i)31-s + (0.809 + 0.587i)35-s + (−0.309 − 0.951i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $0.935 + 0.352i$
Analytic conductor: \(4.69970\)
Root analytic conductor: \(4.69970\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (735, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1012,\ (0:\ ),\ 0.935 + 0.352i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.547905865 + 0.2816068452i\)
\(L(\frac12)\) \(\approx\) \(1.547905865 + 0.2816068452i\)
\(L(1)\) \(\approx\) \(1.125985361 + 0.2084410261i\)
\(L(1)\) \(\approx\) \(1.125985361 + 0.2084410261i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 - T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−21.51847320736123761611881269624, −20.90989892582296028294573545712, −19.85802037822316122112289052420, −19.09116859667257645165905067249, −18.49344830341154308086768575420, −17.58534171884257354203353642564, −17.044282800852564624297046984301, −16.52637537149167015384704878458, −14.951865720479388504635896657315, −14.24338125322603714042461820963, −13.74418356767625071014205977464, −12.95303173335678366703737808416, −11.97466108416147556177880141938, −11.30318741741615012823979074970, −10.212939181059501883771414086712, −9.838399653573745884008137582112, −8.33595852470234675988703563068, −7.600840751057214556921771745369, −6.856877349190126776417969399839, −6.107391344839681178667784453071, −5.2665773095184636991617623007, −4.08448934958080583324864173389, −2.84115222735003761589625047471, −1.90314391530550274998341042502, −1.04607785874390698389223541814, 0.86110160682589316214284622812, 2.36357514249975231606337522467, 3.07893704616334326704665924227, 4.503834144146568963977746198501, 5.31790422876224940106915766603, 5.54909692254242436696610603512, 6.79475428614424615549659553610, 8.09853441947867130543513672799, 9.034778309487139666344323508440, 9.48810939271251238464778090580, 10.322936543009509018980908758579, 11.20951710960730593607819585239, 12.22193069819276275124735745805, 12.64214422409416363011698886993, 14.07101521357030013552813423910, 14.48062098686377360307551276698, 15.65720952270197296516826503604, 15.966159872368217925179583046887, 17.11287024711346402664519766497, 17.567720453340383269244004993089, 18.29474089811996733902623964502, 19.499195548864843794129192433100, 20.33548997711102265837222242655, 21.072482957857512421657929920979, 21.61807867467349716567949917973

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