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Properties

Label 2-2535-195.134-c0-0-10
Degree $2$
Conductor $2535$
Sign $0.327 - 0.944i$
Analytic cond. $1.26512$
Root an. cond. $1.12477$
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  + (1.56 + 0.900i)2-s + (0.5 − 0.866i)3-s + (1.12 + 1.94i)4-s + i·5-s + (1.56 − 0.900i)6-s + 2.24i·8-s + (−0.499 − 0.866i)9-s + (−0.900 + 1.56i)10-s + 2.24·12-s + (0.866 + 0.5i)15-s + (−0.900 + 1.56i)16-s + (0.222 + 0.385i)17-s − 1.80i·18-s + (−1.07 + 0.623i)19-s + (−1.94 + 1.12i)20-s + ⋯
L(s)  = 1  + (1.56 + 0.900i)2-s + (0.5 − 0.866i)3-s + (1.12 + 1.94i)4-s + i·5-s + (1.56 − 0.900i)6-s + 2.24i·8-s + (−0.499 − 0.866i)9-s + (−0.900 + 1.56i)10-s + 2.24·12-s + (0.866 + 0.5i)15-s + (−0.900 + 1.56i)16-s + (0.222 + 0.385i)17-s − 1.80i·18-s + (−1.07 + 0.623i)19-s + (−1.94 + 1.12i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2535\)    =    \(3 \cdot 5 \cdot 13^{2}\)
Sign: $0.327 - 0.944i$
Analytic conductor: \(1.26512\)
Root analytic conductor: \(1.12477\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2535} (1499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2535,\ (\ :0),\ 0.327 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.278467696\)
\(L(\frac12)\) \(\approx\) \(3.278467696\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 - iT \)
13 \( 1 \)
good2 \( 1 + (-1.56 - 0.900i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (1.07 - 0.623i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - 0.445iT - T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + 1.24iT - T^{2} \)
53 \( 1 + 1.24T + T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 1.80T + T^{2} \)
83 \( 1 + 0.445iT - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.706700284443678766236081900475, −8.187498386298437924593387452267, −7.33242091670266134106258201929, −6.77510346798110705771680809619, −6.30328907184615944337820807024, −5.58563422725449299715235818751, −4.43796126129889549453156877059, −3.60516382078328376127687588848, −2.88736468089329168114009308675, −2.06122084282426561174657035982, 1.47252384338743513417152709301, 2.54833876741505153442290336829, 3.36268707713085772526981946660, 4.22295795079986366387956986862, 4.73361791007939664673627530570, 5.39917851830631318591201676234, 6.09688227819872814442739819135, 7.36900576300211002351751281138, 8.354748917934127421769495062955, 9.295780999084855105791809397818

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