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Properties

Label 1-48e2-2304.1397-r1-0-0
Degree $1$
Conductor $2304$
Sign $-0.856 + 0.516i$
Analytic cond. $247.599$
Root an. cond. $247.599$
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  + (−0.528 − 0.849i)5-s + (0.946 − 0.321i)7-s + (−0.935 − 0.352i)11-s + (−0.0327 + 0.999i)13-s + (−0.382 − 0.923i)17-s + (0.956 + 0.290i)19-s + (−0.896 + 0.442i)23-s + (−0.442 + 0.896i)25-s + (0.986 + 0.162i)29-s + (0.965 − 0.258i)31-s + (−0.773 − 0.634i)35-s + (−0.290 − 0.956i)37-s + (0.442 + 0.896i)41-s + (0.412 + 0.910i)43-s + (−0.793 − 0.608i)47-s + ⋯
L(s)  = 1  + (−0.528 − 0.849i)5-s + (0.946 − 0.321i)7-s + (−0.935 − 0.352i)11-s + (−0.0327 + 0.999i)13-s + (−0.382 − 0.923i)17-s + (0.956 + 0.290i)19-s + (−0.896 + 0.442i)23-s + (−0.442 + 0.896i)25-s + (0.986 + 0.162i)29-s + (0.965 − 0.258i)31-s + (−0.773 − 0.634i)35-s + (−0.290 − 0.956i)37-s + (0.442 + 0.896i)41-s + (0.412 + 0.910i)43-s + (−0.793 − 0.608i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-0.856 + 0.516i$
Analytic conductor: \(247.599\)
Root analytic conductor: \(247.599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2304,\ (1:\ ),\ -0.856 + 0.516i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06085448454 - 0.2187391029i\)
\(L(\frac12)\) \(\approx\) \(-0.06085448454 - 0.2187391029i\)
\(L(1)\) \(\approx\) \(0.8907706989 - 0.1929662400i\)
\(L(1)\) \(\approx\) \(0.8907706989 - 0.1929662400i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.528 - 0.849i)T \)
7 \( 1 + (0.946 - 0.321i)T \)
11 \( 1 + (-0.935 - 0.352i)T \)
13 \( 1 + (-0.0327 + 0.999i)T \)
17 \( 1 + (-0.382 - 0.923i)T \)
19 \( 1 + (0.956 + 0.290i)T \)
23 \( 1 + (-0.896 + 0.442i)T \)
29 \( 1 + (0.986 + 0.162i)T \)
31 \( 1 + (0.965 - 0.258i)T \)
37 \( 1 + (-0.290 - 0.956i)T \)
41 \( 1 + (0.442 + 0.896i)T \)
43 \( 1 + (0.412 + 0.910i)T \)
47 \( 1 + (-0.793 - 0.608i)T \)
53 \( 1 + (-0.634 - 0.773i)T \)
59 \( 1 + (0.849 - 0.528i)T \)
61 \( 1 + (-0.812 + 0.582i)T \)
67 \( 1 + (-0.910 - 0.412i)T \)
71 \( 1 + (0.980 + 0.195i)T \)
73 \( 1 + (-0.195 - 0.980i)T \)
79 \( 1 + (-0.130 - 0.991i)T \)
83 \( 1 + (-0.973 + 0.227i)T \)
89 \( 1 + (0.831 - 0.555i)T \)
97 \( 1 + (0.258 - 0.965i)T \)
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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

−19.94248401284366077235317792128, −19.05570362177153848483400245889, −18.37795736119430672263481322051, −17.72898626944701010963112411806, −17.39433520287049626072215807905, −15.883114844909559235317743362064, −15.58750040848520738286703235921, −14.955018129748969751225805121903, −14.16564029039769526762694041006, −13.51303448176989394026354204798, −12.39160125057113775860504823980, −11.970288379294015383374323612658, −10.9661692238071561893833277436, −10.518132065617179101691686859181, −9.84881655829074653401347458332, −8.4769801824477186474173859119, −8.06331598139112410733233258040, −7.43049990276038233542745152901, −6.47321307147075869764381170264, −5.60386116501684767804881717237, −4.80775152882489317814420089547, −3.99242765631275318964215050455, −2.885391883261380978445608606288, −2.40116616892352093751082391802, −1.1825412065200982266609503267, 0.04274757727408051975887866538, 0.99400296292438084204299185235, 1.85203766932688196561763407377, 2.92983464665555281573126138063, 4.00926704570573806114790963627, 4.7465800198290312618815879486, 5.205224854802894646088252236293, 6.27700362130493849882138349273, 7.42539207883802867455607783706, 7.87548774175583955224885972860, 8.58351982553916886053139127308, 9.41549799099370951338981011926, 10.19730089320272985602423416819, 11.33007537534249138060982283989, 11.592513985558374959974852838344, 12.3778032550051528420234191972, 13.40812325182732037078508834690, 13.877168785174901252352345403372, 14.61823823268951996459590065300, 15.75309259484416558976509507660, 16.07797255273654770078420585720, 16.72263800491599021951315381695, 17.841125309449840147486008793024, 18.06437589075656451676410816796, 19.14946947601059321567164478488

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