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Properties

Label 1-2300-2300.91-r0-0-0
Degree $1$
Conductor $2300$
Sign $0.728 - 0.684i$
Analytic cond. $10.6811$
Root an. cond. $10.6811$
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  + (0.809 − 0.587i)3-s + 7-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (0.309 − 0.951i)13-s + (0.809 + 0.587i)17-s + (−0.809 − 0.587i)19-s + (0.809 − 0.587i)21-s + (−0.309 − 0.951i)27-s + (−0.809 + 0.587i)29-s + (0.809 + 0.587i)31-s + (0.809 + 0.587i)33-s + (−0.309 + 0.951i)37-s + (−0.309 − 0.951i)39-s + (0.309 − 0.951i)41-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)3-s + 7-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (0.309 − 0.951i)13-s + (0.809 + 0.587i)17-s + (−0.809 − 0.587i)19-s + (0.809 − 0.587i)21-s + (−0.309 − 0.951i)27-s + (−0.809 + 0.587i)29-s + (0.809 + 0.587i)31-s + (0.809 + 0.587i)33-s + (−0.309 + 0.951i)37-s + (−0.309 − 0.951i)39-s + (0.309 − 0.951i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2300\)    =    \(2^{2} \cdot 5^{2} \cdot 23\)
Sign: $0.728 - 0.684i$
Analytic conductor: \(10.6811\)
Root analytic conductor: \(10.6811\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2300} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2300,\ (0:\ ),\ 0.728 - 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.599760707 - 1.029318080i\)
\(L(\frac12)\) \(\approx\) \(2.599760707 - 1.029318080i\)
\(L(1)\) \(\approx\) \(1.614589178 - 0.3601929909i\)
\(L(1)\) \(\approx\) \(1.614589178 - 0.3601929909i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 + T \)
11 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
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 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
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 \( 1 + (-0.309 - 0.951i)T \)
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

−19.71843361045256528719287604611, −18.87520136489391643290901913590, −18.692375261891252562591271342665, −17.45789315032742205932526359242, −16.6707958336432292141619880750, −16.227074702682553536071656320188, −15.273329322727218994620360226407, −14.59697458548710417677185134256, −14.022113143029656471036591604507, −13.59675904178402240166152840323, −12.453430360524863670347041608332, −11.495464769522509278461085892675, −11.02301770414961885071179686767, −10.17172201647348794507739559492, −9.264068692769794204145604563495, −8.74205721545344287717848240762, −7.961376666175301316202642850399, −7.39734293698343362688440142273, −6.13478323108703024037325329101, −5.41830726280038669157527058670, −4.289528994331763991715682766956, −4.00209570855884164685740291226, −2.875289546418424859666683403706, −2.05138284713397093830538121876, −1.125237858176351596974871109993, 0.97443160380484181083276328532, 1.74845128022704684654500761555, 2.51848408151514722917762385999, 3.522644876165265965525076562329, 4.31406983440922154984415768325, 5.24247685613249206741846435607, 6.17831527125921104559356386568, 7.15901593266117507360565768506, 7.67034691983886274984099258153, 8.48397780322920477044471646278, 8.96709970373575429260120943842, 10.07582264881971964989399330261, 10.66501346731064264482436476316, 11.722409411242755712128645511696, 12.44437572598057725091973353652, 12.955069666901752778685987191528, 13.89771701245079121969064750273, 14.4892709774493055556567040965, 15.18494990991359139884400940993, 15.54704691770832916447117543081, 17.07604350618170847514924354840, 17.39384968493441267878265018257, 18.2197347045207693735912945114, 18.75268990834533634796310799137, 19.73305753483438846918108922216

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