Properties

Label 2-69-69.53-c1-0-4
Degree $2$
Conductor $69$
Sign $-0.637 + 0.770i$
Analytic cond. $0.550967$
Root an. cond. $0.742272$
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  + (−0.924 − 0.801i)2-s + (−1.53 + 0.798i)3-s + (−0.0715 − 0.497i)4-s + (−1.44 − 3.17i)5-s + (2.06 + 0.493i)6-s + (−0.466 − 1.58i)7-s + (−1.65 + 2.57i)8-s + (1.72 − 2.45i)9-s + (−1.20 + 4.09i)10-s + (2.29 + 2.65i)11-s + (0.507 + 0.707i)12-s + (−1.83 − 0.538i)13-s + (−0.841 + 1.84i)14-s + (4.75 + 3.71i)15-s + (2.63 − 0.772i)16-s + (0.468 − 3.25i)17-s + ⋯
L(s)  = 1  + (−0.653 − 0.566i)2-s + (−0.887 + 0.460i)3-s + (−0.0357 − 0.248i)4-s + (−0.647 − 1.41i)5-s + (0.841 + 0.201i)6-s + (−0.176 − 0.600i)7-s + (−0.585 + 0.910i)8-s + (0.575 − 0.818i)9-s + (−0.379 + 1.29i)10-s + (0.693 + 0.799i)11-s + (0.146 + 0.204i)12-s + (−0.508 − 0.149i)13-s + (−0.224 + 0.492i)14-s + (1.22 + 0.959i)15-s + (0.657 − 0.193i)16-s + (0.113 − 0.790i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.637 + 0.770i$
Analytic conductor: \(0.550967\)
Root analytic conductor: \(0.742272\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :1/2),\ -0.637 + 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.174700 - 0.371179i\)
\(L(\frac12)\) \(\approx\) \(0.174700 - 0.371179i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.53 - 0.798i)T \)
23 \( 1 + (-3.15 + 3.61i)T \)
good2 \( 1 + (0.924 + 0.801i)T + (0.284 + 1.97i)T^{2} \)
5 \( 1 + (1.44 + 3.17i)T + (-3.27 + 3.77i)T^{2} \)
7 \( 1 + (0.466 + 1.58i)T + (-5.88 + 3.78i)T^{2} \)
11 \( 1 + (-2.29 - 2.65i)T + (-1.56 + 10.8i)T^{2} \)
13 \( 1 + (1.83 + 0.538i)T + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (-0.468 + 3.25i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (-4.45 + 0.640i)T + (18.2 - 5.35i)T^{2} \)
29 \( 1 + (2.57 + 0.369i)T + (27.8 + 8.17i)T^{2} \)
31 \( 1 + (6.47 + 4.16i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (0.285 + 0.130i)T + (24.2 + 27.9i)T^{2} \)
41 \( 1 + (-10.0 + 4.59i)T + (26.8 - 30.9i)T^{2} \)
43 \( 1 + (-2.84 - 4.43i)T + (-17.8 + 39.1i)T^{2} \)
47 \( 1 + 0.189iT - 47T^{2} \)
53 \( 1 + (6.53 - 1.91i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (1.58 - 5.38i)T + (-49.6 - 31.8i)T^{2} \)
61 \( 1 + (0.426 - 0.664i)T + (-25.3 - 55.4i)T^{2} \)
67 \( 1 + (-5.69 - 4.93i)T + (9.53 + 66.3i)T^{2} \)
71 \( 1 + (-3.14 - 2.72i)T + (10.1 + 70.2i)T^{2} \)
73 \( 1 + (-0.0940 - 0.653i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (2.86 - 9.76i)T + (-66.4 - 42.7i)T^{2} \)
83 \( 1 + (-1.89 + 4.14i)T + (-54.3 - 62.7i)T^{2} \)
89 \( 1 + (7.01 - 4.50i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (-17.0 + 7.79i)T + (63.5 - 73.3i)T^{2} \)
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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

−14.53046329544235822193030633411, −12.76079855314563911895428506604, −11.93048951290723747146782952475, −11.03712846500446963117629554204, −9.687239160981083568894574201924, −9.147499069833682542842307585634, −7.35708126957873668368351685646, −5.40650285624266335542525906395, −4.34150237429391038043791834342, −0.813943230355035838806824932887, 3.42746624490387285109086438843, 5.94660672266318195229988178617, 6.99026203715213283963887827462, 7.77898850307370628776998494667, 9.345525837601019476180422182145, 10.85994419766739782995787111656, 11.71506415289939190677440975160, 12.69579373505850442704190818197, 14.28324925270956893935285905650, 15.43834519647302744879799386371

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