Properties

Label 2-69-23.4-c1-0-1
Degree $2$
Conductor $69$
Sign $0.268 - 0.963i$
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  + (0.915 + 2.00i)2-s + (0.959 − 0.281i)3-s + (−1.87 + 2.15i)4-s + (−2.61 − 1.68i)5-s + (1.44 + 1.66i)6-s + (−0.427 − 2.97i)7-s + (−1.80 − 0.531i)8-s + (0.841 − 0.540i)9-s + (0.976 − 6.79i)10-s + (−2.48 + 5.43i)11-s + (−1.18 + 2.59i)12-s + (−0.0566 + 0.393i)13-s + (5.57 − 3.58i)14-s + (−2.98 − 0.877i)15-s + (0.221 + 1.53i)16-s + (0.862 + 0.995i)17-s + ⋯
L(s)  = 1  + (0.647 + 1.41i)2-s + (0.553 − 0.162i)3-s + (−0.935 + 1.07i)4-s + (−1.17 − 0.752i)5-s + (0.589 + 0.679i)6-s + (−0.161 − 1.12i)7-s + (−0.639 − 0.187i)8-s + (0.280 − 0.180i)9-s + (0.308 − 2.14i)10-s + (−0.748 + 1.63i)11-s + (−0.342 + 0.749i)12-s + (−0.0157 + 0.109i)13-s + (1.48 − 0.957i)14-s + (−0.771 − 0.226i)15-s + (0.0553 + 0.384i)16-s + (0.209 + 0.241i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.268 - 0.963i$
Analytic conductor: \(0.550967\)
Root analytic conductor: \(0.742272\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :1/2),\ 0.268 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.945399 + 0.717563i\)
\(L(\frac12)\) \(\approx\) \(0.945399 + 0.717563i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.959 + 0.281i)T \)
23 \( 1 + (-2.43 + 4.13i)T \)
good2 \( 1 + (-0.915 - 2.00i)T + (-1.30 + 1.51i)T^{2} \)
5 \( 1 + (2.61 + 1.68i)T + (2.07 + 4.54i)T^{2} \)
7 \( 1 + (0.427 + 2.97i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (2.48 - 5.43i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (0.0566 - 0.393i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (-0.862 - 0.995i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (-1.67 + 1.93i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (0.836 + 0.965i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-0.575 - 0.169i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (6.75 - 4.34i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (7.15 + 4.59i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (-7.26 + 2.13i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 - 7.68T + 47T^{2} \)
53 \( 1 + (-0.399 - 2.78i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-0.477 + 3.32i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (-0.182 - 0.0536i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (-3.73 - 8.18i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (5.61 + 12.3i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (9.37 - 10.8i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (0.718 - 4.99i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (-6.86 + 4.41i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (0.416 - 0.122i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (-0.265 - 0.170i)T + (40.2 + 88.2i)T^{2} \)
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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

−15.21997630880887204884130058526, −14.06258906747576780174392316339, −13.06644580798020493499830101159, −12.27460053660880516049586122447, −10.35055630441435860270506728818, −8.640380827891177972968687855733, −7.52921500319124319788946549231, −7.07268702928316015884821927294, −4.88740516858702272645932748937, −4.03423105257628206892948813824, 2.85783205743221049117445864961, 3.57112208244166780267811491598, 5.46312092844551907230211753743, 7.68300750258443462462951847357, 8.972629295530091978883143878345, 10.47781032179285153928788860904, 11.34793909614047840511799135882, 12.11067926355913698118373085325, 13.30921091504589360407192390936, 14.28814307254460397590581478039

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