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Properties

Label 1-304-304.21-r1-0-0
Degree $1$
Conductor $304$
Sign $0.721 - 0.692i$
Analytic cond. $32.6693$
Root an. cond. $32.6693$
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  + (−0.984 + 0.173i)3-s + (0.642 + 0.766i)5-s + (0.5 − 0.866i)7-s + (0.939 − 0.342i)9-s + (0.866 − 0.5i)11-s + (0.984 + 0.173i)13-s + (−0.766 − 0.642i)15-s + (−0.939 − 0.342i)17-s + (−0.342 + 0.939i)21-s + (−0.766 − 0.642i)23-s + (−0.173 + 0.984i)25-s + (−0.866 + 0.5i)27-s + (−0.342 − 0.939i)29-s + (0.5 − 0.866i)31-s + (−0.766 + 0.642i)33-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)3-s + (0.642 + 0.766i)5-s + (0.5 − 0.866i)7-s + (0.939 − 0.342i)9-s + (0.866 − 0.5i)11-s + (0.984 + 0.173i)13-s + (−0.766 − 0.642i)15-s + (−0.939 − 0.342i)17-s + (−0.342 + 0.939i)21-s + (−0.766 − 0.642i)23-s + (−0.173 + 0.984i)25-s + (−0.866 + 0.5i)27-s + (−0.342 − 0.939i)29-s + (0.5 − 0.866i)31-s + (−0.766 + 0.642i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.721 - 0.692i$
Analytic conductor: \(32.6693\)
Root analytic conductor: \(32.6693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 304,\ (1:\ ),\ 0.721 - 0.692i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.548476414 - 0.6229135562i\)
\(L(\frac12)\) \(\approx\) \(1.548476414 - 0.6229135562i\)
\(L(1)\) \(\approx\) \(1.026359456 - 0.06849166464i\)
\(L(1)\) \(\approx\) \(1.026359456 - 0.06849166464i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.984 + 0.173i)T \)
5 \( 1 + (0.642 + 0.766i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (0.984 + 0.173i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (-0.342 - 0.939i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (0.642 + 0.766i)T \)
47 \( 1 + (-0.939 + 0.342i)T \)
53 \( 1 + (0.642 - 0.766i)T \)
59 \( 1 + (-0.342 + 0.939i)T \)
61 \( 1 + (0.642 - 0.766i)T \)
67 \( 1 + (-0.342 - 0.939i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (-0.173 - 0.984i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (0.939 + 0.342i)T \)
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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

−25.01918094047622157237997177656, −24.31415989895529944020797228096, −23.55850461552700359898220286784, −22.33494183437616002084891617608, −21.79249035165268314544526926112, −20.8892244307609584104113557009, −19.88047466639764010588949568915, −18.58865771098438258289000329499, −17.73742513571699398504133942807, −17.30440085530921347008459642119, −16.14154082274383822691983562061, −15.40677298160905581665766158883, −14.037244371898194197843772429592, −13.01207866216332520242908642440, −12.19696149745318152448611416454, −11.43258212173061745791495981952, −10.32910567092228527946810944355, −9.17365194340905230814447342300, −8.368785323703224837210741920145, −6.81073962505541886626612053795, −5.912039510355178994393429181254, −5.12689576304317175143840211809, −4.07118574688198925134028345691, −1.9856217228474574073718701098, −1.197986217534974881875304120338, 0.644481913465391332886727448722, 1.91141587190684508215668740465, 3.70109579497460661636564590164, 4.58271649133001331429106625812, 6.11230861582506044938291756936, 6.46472510075763872239422570501, 7.709435919547566360748691199845, 9.2211640702512957952769167325, 10.23049939056099571792601063628, 11.13098713977534566731114929081, 11.539609843665181468182087001012, 13.14736521875514134809183159744, 13.8851580336743255534311822230, 14.86072693565840276224777911917, 16.08815881782792458459991638016, 16.899113819590045076667398206095, 17.76477210649289000079839412473, 18.3125015023016676920687553050, 19.498772090589286359662592279365, 20.78998159602240046520814236018, 21.46285074817776982169461475657, 22.53228319299569383734268631122, 22.89439641415982143518591520422, 24.111616426938943787328650955535, 24.7457559869746225245410795395

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