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Properties

Label 2-245-5.4-c1-0-12
Degree $2$
Conductor $245$
Sign $-0.894 + 0.447i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
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  − 2i·2-s − i·3-s − 2·4-s + (2 − i)5-s − 2·6-s + 2·9-s + (−2 − 4i)10-s − 3·11-s + 2i·12-s − i·13-s + (−1 − 2i)15-s − 4·16-s + 7i·17-s − 4i·18-s + (−4 + 2i)20-s + ⋯
L(s)  = 1  − 1.41i·2-s − 0.577i·3-s − 4-s + (0.894 − 0.447i)5-s − 0.816·6-s + 0.666·9-s + (−0.632 − 1.26i)10-s − 0.904·11-s + 0.577i·12-s − 0.277i·13-s + (−0.258 − 0.516i)15-s − 16-s + 1.69i·17-s − 0.942i·18-s + (−0.894 + 0.447i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.322312 - 1.36533i\)
\(L(\frac12)\) \(\approx\) \(0.322312 - 1.36533i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2 + i)T \)
7 \( 1 \)
good2 \( 1 + 2iT - 2T^{2} \)
3 \( 1 + iT - 3T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 - 7iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 5T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 7iT - 97T^{2} \)
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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

−11.84499212526274276054351731312, −10.48830894694029595254842704192, −10.21239243694715159637043713959, −9.083276566823832879511463172761, −7.919503696406930494753125122485, −6.57030394030377988870854436187, −5.31323077445712023937989250955, −3.88899238689639897784968487792, −2.34216171330090280540722742525, −1.33621710577323121618905087609, 2.62782325971648773735830990284, 4.64975090428334336306619623857, 5.38402049899582686739742874953, 6.61974670488315296105460364924, 7.25555219626301205548661667912, 8.497055539128846024946525050910, 9.552653034685264689496698174365, 10.28435667535972885534450575044, 11.35436837004902089596449340197, 12.87800358646590012494933048348

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