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Properties

Label 1-1148-1148.975-r1-0-0
Degree $1$
Conductor $1148$
Sign $0.270 + 0.962i$
Analytic cond. $123.369$
Root an. cond. $123.369$
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  + (−0.866 − 0.5i)3-s + (0.5 + 0.866i)5-s + (0.5 + 0.866i)9-s + (−0.866 − 0.5i)11-s − i·13-s − i·15-s + (−0.866 − 0.5i)17-s + (−0.866 + 0.5i)19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − i·27-s − i·29-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)33-s + (−0.5 − 0.866i)37-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (0.5 + 0.866i)5-s + (0.5 + 0.866i)9-s + (−0.866 − 0.5i)11-s − i·13-s − i·15-s + (−0.866 − 0.5i)17-s + (−0.866 + 0.5i)19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − i·27-s − i·29-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)33-s + (−0.5 − 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.270 + 0.962i$
Analytic conductor: \(123.369\)
Root analytic conductor: \(123.369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (975, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (1:\ ),\ 0.270 + 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6504341560 + 0.4928233180i\)
\(L(\frac12)\) \(\approx\) \(0.6504341560 + 0.4928233180i\)
\(L(1)\) \(\approx\) \(0.7514590300 + 0.01858039448i\)
\(L(1)\) \(\approx\) \(0.7514590300 + 0.01858039448i\)

Euler product

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

−21.20681057360768734732520730661, −20.38823103243019485874818099787, −19.47870739436688931765754520280, −18.38847906360851185313682501474, −17.71198718962769164337370784987, −17.03544390291757788072346275630, −16.41257876617749153160258213029, −15.67396244757336179593783496928, −14.92552203615010741264163765285, −13.82352256142167330102223701687, −12.816167126150948876806849452407, −12.49999321078282283922633348619, −11.414263772478886643580637523716, −10.619611978391240070857678033546, −9.9576024179544646989359410935, −8.98879983536721961194644331038, −8.44835604663071911301469347269, −6.89331167047045243847697395369, −6.42787977139594478087118329029, −5.21532129992017800140202357383, −4.77059163201384568294604544657, −4.00059150441868385951491235278, −2.45905938289535077467571411289, −1.46637144546918807906812566654, −0.25377464643606192110164771818, 0.728570993302221568036750477407, 2.11489090216961867085514496072, 2.76526457035823633295203290302, 4.06615475395053003163930328443, 5.33638250977575484136750154098, 5.82421510428828866264577102279, 6.66192436385830173832538751387, 7.509012523586315339812621070821, 8.22509445717869980748837305351, 9.58549693927861394526051829438, 10.43648128204389242690405593076, 10.96691326937510506629370966309, 11.64499454424119036471038453633, 12.8641208820994537893715960699, 13.269254362451877052357097828221, 14.07154818740025216355723167008, 15.24184838554513387177640104155, 15.750881285003891862949545867814, 16.82800704680167790622758114575, 17.68494215578123566758365533263, 17.93924040843254584992992184707, 18.96212506443011564104669539889, 19.311765402018180331439629561410, 20.7102334823852044148523716994, 21.36435825751911036976629415249

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