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Properties

Label 2-1575-21.5-c1-0-0
Degree $2$
Conductor $1575$
Sign $-0.366 - 0.930i$
Analytic cond. $12.5764$
Root an. cond. $3.54632$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.14 − 0.659i)2-s + (−0.130 − 0.226i)4-s + (−1.57 + 2.12i)7-s + 2.98i·8-s + (2.08 − 1.20i)11-s − 1.69i·13-s + (3.19 − 1.38i)14-s + (1.70 − 2.95i)16-s + (0.480 + 0.831i)17-s + (−3.56 − 2.05i)19-s − 3.17·22-s + (4.99 + 2.88i)23-s + (−1.11 + 1.93i)26-s + (0.688 + 0.0786i)28-s − 5.56i·29-s + ⋯
L(s)  = 1  + (−0.807 − 0.466i)2-s + (−0.0654 − 0.113i)4-s + (−0.595 + 0.803i)7-s + 1.05i·8-s + (0.629 − 0.363i)11-s − 0.469i·13-s + (0.855 − 0.371i)14-s + (0.425 − 0.737i)16-s + (0.116 + 0.201i)17-s + (−0.818 − 0.472i)19-s − 0.677·22-s + (1.04 + 0.601i)23-s + (−0.218 + 0.379i)26-s + (0.130 + 0.0148i)28-s − 1.03i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.366 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.366 - 0.930i$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ -0.366 - 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2489058895\)
\(L(\frac12)\) \(\approx\) \(0.2489058895\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 + (1.57 - 2.12i)T \)
good2 \( 1 + (1.14 + 0.659i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-2.08 + 1.20i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.69iT - 13T^{2} \)
17 \( 1 + (-0.480 - 0.831i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.56 + 2.05i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.99 - 2.88i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.56iT - 29T^{2} \)
31 \( 1 + (7.58 - 4.37i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.98 + 3.44i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.87T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + (5.05 - 8.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (11.0 - 6.35i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.38 - 5.86i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.98 - 1.14i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.39 - 4.15i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.24iT - 71T^{2} \)
73 \( 1 + (12.7 - 7.34i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.892 - 1.54i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.62T + 83T^{2} \)
89 \( 1 + (-0.220 + 0.382i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.4iT - 97T^{2} \)
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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

−9.558889496232354992865804124057, −8.983073232779610228818718299590, −8.468033295895331832878237763257, −7.45350634208263347676791737588, −6.34716941102894244317182585965, −5.68630326072068933368431275154, −4.77412156287273694399584148315, −3.43538170527945180415433799117, −2.49348465815594991516640662894, −1.32490909151959943647671593671, 0.13758186125237994350181838941, 1.58890020707008276256449650951, 3.29851539059329412306490935648, 3.99380044920686682852718170396, 4.96815119449603213195440327368, 6.52791986711176719175417451520, 6.72768644670309310929469627988, 7.59176831415267985319752637785, 8.418699864000399403135432293006, 9.183984605591919677579487258362

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