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Properties

Label 1-1575-1575.1361-r0-0-0
Degree $1$
Conductor $1575$
Sign $0.879 + 0.476i$
Analytic cond. $7.31426$
Root an. cond. $7.31426$
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)2-s + (−0.809 − 0.587i)4-s + (0.809 − 0.587i)8-s + (−0.669 − 0.743i)11-s + (0.978 + 0.207i)13-s + (0.309 + 0.951i)16-s + (0.913 + 0.406i)17-s + (0.104 + 0.994i)19-s + (0.913 − 0.406i)22-s + (0.978 − 0.207i)23-s + (−0.5 + 0.866i)26-s + (0.104 − 0.994i)29-s + (0.809 − 0.587i)31-s − 32-s + (−0.669 + 0.743i)34-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.809 − 0.587i)4-s + (0.809 − 0.587i)8-s + (−0.669 − 0.743i)11-s + (0.978 + 0.207i)13-s + (0.309 + 0.951i)16-s + (0.913 + 0.406i)17-s + (0.104 + 0.994i)19-s + (0.913 − 0.406i)22-s + (0.978 − 0.207i)23-s + (−0.5 + 0.866i)26-s + (0.104 − 0.994i)29-s + (0.809 − 0.587i)31-s − 32-s + (−0.669 + 0.743i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.879 + 0.476i$
Analytic conductor: \(7.31426\)
Root analytic conductor: \(7.31426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1575,\ (0:\ ),\ 0.879 + 0.476i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.189262909 + 0.3016174322i\)
\(L(\frac12)\) \(\approx\) \(1.189262909 + 0.3016174322i\)
\(L(1)\) \(\approx\) \(0.8598412178 + 0.2854227016i\)
\(L(1)\) \(\approx\) \(0.8598412178 + 0.2854227016i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-0.669 - 0.743i)T \)
13 \( 1 + (0.978 + 0.207i)T \)
17 \( 1 + (0.913 + 0.406i)T \)
19 \( 1 + (0.104 + 0.994i)T \)
23 \( 1 + (0.978 - 0.207i)T \)
29 \( 1 + (0.104 - 0.994i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.978 - 0.207i)T \)
41 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (0.104 - 0.994i)T \)
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 \( 1 + (0.978 - 0.207i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (0.913 + 0.406i)T \)
89 \( 1 + (0.669 + 0.743i)T \)
97 \( 1 + (0.104 - 0.994i)T \)
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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

−20.444683409848293822376868690366, −19.81650258916283064677961012196, −18.92163964226439202302647824921, −18.31821941305919062267001058349, −17.70204459470919074027758042790, −16.94646057302834913229965959697, −16.00797787313185099009323107892, −15.270019529245007731490392757852, −14.206278563256105315376544644243, −13.46046079880171826764816167480, −12.81152993362886921911971712042, −12.09729575638842866929819528415, −11.21955352460315217918384571031, −10.59788683003254663765771561304, −9.83906549367871653712810909337, −9.06137567549610603077308556284, −8.27651638818373414852511758523, −7.48350382585282625282733341745, −6.55997867860859986438733889932, −5.09068835220429383069351212759, −4.82319172359511953184104481613, −3.384040717841530724028036364208, −2.99651143160011362675625201538, −1.77867516127723393088586010376, −0.91566170778796041448929640043, 0.690625505884672301098272393882, 1.74191303871972062826941895836, 3.25382821242368353930947788735, 4.0066021811424875294428499707, 5.17931047520669004531080366060, 5.786328453411581206931989461188, 6.518337903846744644770712357956, 7.47139182113335213023860481921, 8.359602713697429388732124555771, 8.63879310652819950148283429318, 9.913700649489701108300469711731, 10.3709278964000211103868310605, 11.34003975515681277665462659511, 12.357243778240996339932736948678, 13.47532424398940257136580065331, 13.68203604480902019250913918295, 14.775335552170876543278145257944, 15.35079909526465584200887802627, 16.237343683542385192236117108778, 16.67711938650848217559295402752, 17.46724257951171857564231034951, 18.4481334627464052618130845247, 18.836692485910465913504014108788, 19.44640261976167376136162123947, 20.88150630640902794151776737563

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