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Properties

Label 1-1575-1575.1129-r0-0-0
Degree $1$
Conductor $1575$
Sign $-0.615 - 0.788i$
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  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)11-s + (−0.913 − 0.406i)13-s + (−0.809 + 0.587i)16-s + (−0.669 − 0.743i)17-s + (−0.978 + 0.207i)19-s + (−0.669 + 0.743i)22-s + (−0.913 + 0.406i)23-s + (−0.5 − 0.866i)26-s + (−0.978 − 0.207i)29-s + (0.309 − 0.951i)31-s − 32-s + (−0.104 − 0.994i)34-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.309 + 0.951i)4-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)11-s + (−0.913 − 0.406i)13-s + (−0.809 + 0.587i)16-s + (−0.669 − 0.743i)17-s + (−0.978 + 0.207i)19-s + (−0.669 + 0.743i)22-s + (−0.913 + 0.406i)23-s + (−0.5 − 0.866i)26-s + (−0.978 − 0.207i)29-s + (0.309 − 0.951i)31-s − 32-s + (−0.104 − 0.994i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.615 - 0.788i$
Analytic conductor: \(7.31426\)
Root analytic conductor: \(7.31426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1575,\ (0:\ ),\ -0.615 - 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1693851258 + 0.3470516440i\)
\(L(\frac12)\) \(\approx\) \(-0.1693851258 + 0.3470516440i\)
\(L(1)\) \(\approx\) \(1.003621457 + 0.5408161029i\)
\(L(1)\) \(\approx\) \(1.003621457 + 0.5408161029i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (-0.913 - 0.406i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (-0.913 + 0.406i)T \)
29 \( 1 + (-0.978 - 0.207i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (-0.913 - 0.406i)T \)
41 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (0.978 + 0.207i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.913 + 0.406i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.669 - 0.743i)T \)
89 \( 1 + (-0.104 + 0.994i)T \)
97 \( 1 + (0.978 + 0.207i)T \)
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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

−19.9069652068713688204351536918, −19.48046489825473764539195143991, −18.85368560169835015520988938776, −17.912840992412143885985670737483, −16.96617955935422109554874704466, −16.1573845219281490799564139104, −15.37437206001111570447187652656, −14.571582227867051424301846390086, −14.01419362640967765915768047386, −13.159666043840257271674615191231, −12.49958191516673866911979370806, −11.77893844802787719646126770837, −10.84673120733418405839260978019, −10.468179635736283899017761628844, −9.36285899843150161256381573916, −8.637029840053543502048982112878, −7.51593736635996671635260428176, −6.44687767034208846032194541165, −5.96354284335553746602390925637, −4.86908701371150915644253457176, −4.21743679465701748581031639983, −3.293443652756385689086998996069, −2.37871609897721504063349295102, −1.57662362184938025028868519398, −0.092630051025440726988474346472, 2.04712912313191915833957818580, 2.553385837206333873471798628379, 3.86625061867546121802059155695, 4.46446371194858163676231070209, 5.331616128540596697701673181584, 6.09369268857658818204735525764, 7.16278023511473889581597673964, 7.50735502913646631718254115448, 8.519663329866226112715302125270, 9.4623664897845531533967389994, 10.33109977155236114636299974554, 11.37227350967086697855152799858, 12.13120488419472624516373538933, 12.75040549984827785037446083936, 13.48268624414777751156137419637, 14.3079949034814366152942868405, 15.10108281486887121561335846223, 15.47649486891952216134071603751, 16.43684594625049333846506540216, 17.25060878974318598044533482739, 17.69901289543299181718011773446, 18.63743818988630437833501689855, 19.8129375840778004541132004174, 20.30288137356499317168162491328, 21.10748103079254423163674944108

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