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Properties

Label 2-245-35.27-c1-0-11
Degree $2$
Conductor $245$
Sign $0.298 + 0.954i$
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  + (−0.366 − 0.366i)2-s + (0.366 + 0.366i)3-s − 1.73i·4-s + (2 − i)5-s − 0.267i·6-s + (−1.36 + 1.36i)8-s − 2.73i·9-s + (−1.09 − 0.366i)10-s − 2.73·11-s + (0.633 − 0.633i)12-s + (2 + 2i)13-s + (1.09 + 0.366i)15-s − 2.46·16-s + (2.73 − 2.73i)17-s + (−1 + i)18-s + 0.732·19-s + ⋯
L(s)  = 1  + (−0.258 − 0.258i)2-s + (0.211 + 0.211i)3-s − 0.866i·4-s + (0.894 − 0.447i)5-s − 0.109i·6-s + (−0.482 + 0.482i)8-s − 0.910i·9-s + (−0.347 − 0.115i)10-s − 0.823·11-s + (0.183 − 0.183i)12-s + (0.554 + 0.554i)13-s + (0.283 + 0.0945i)15-s − 0.616·16-s + (0.662 − 0.662i)17-s + (−0.235 + 0.235i)18-s + 0.167·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.298 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.298 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.298 + 0.954i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 0.298 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02395 - 0.752317i\)
\(L(\frac12)\) \(\approx\) \(1.02395 - 0.752317i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.366 + 0.366i)T + 2iT^{2} \)
3 \( 1 + (-0.366 - 0.366i)T + 3iT^{2} \)
11 \( 1 + 2.73T + 11T^{2} \)
13 \( 1 + (-2 - 2i)T + 13iT^{2} \)
17 \( 1 + (-2.73 + 2.73i)T - 17iT^{2} \)
19 \( 1 - 0.732T + 19T^{2} \)
23 \( 1 + (-0.0980 + 0.0980i)T - 23iT^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 - 7.46iT - 31T^{2} \)
37 \( 1 + (-3.46 - 3.46i)T + 37iT^{2} \)
41 \( 1 - 6.46iT - 41T^{2} \)
43 \( 1 + (-2.83 + 2.83i)T - 43iT^{2} \)
47 \( 1 + (6.46 - 6.46i)T - 47iT^{2} \)
53 \( 1 + (5 - 5i)T - 53iT^{2} \)
59 \( 1 + 8.19T + 59T^{2} \)
61 \( 1 - 1.53iT - 61T^{2} \)
67 \( 1 + (-7.83 - 7.83i)T + 67iT^{2} \)
71 \( 1 - 1.26T + 71T^{2} \)
73 \( 1 + (-9.46 - 9.46i)T + 73iT^{2} \)
79 \( 1 + 3.26iT - 79T^{2} \)
83 \( 1 + (-2.09 - 2.09i)T + 83iT^{2} \)
89 \( 1 + 0.660T + 89T^{2} \)
97 \( 1 + (-5.92 + 5.92i)T - 97iT^{2} \)
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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

−11.79702973495643538979707045732, −10.74419804965027121468669883331, −9.774543563920664967609639222832, −9.350959861925660916315718901782, −8.315101571278314009171164222710, −6.63161469888868005144318806171, −5.74305789483629687200663298285, −4.71063462798713931617043135866, −2.84979098402822344416459075058, −1.23663971919220932039438123941, 2.25563626249003951414373620657, 3.44059216215252064596014701302, 5.23106126221524058469644527512, 6.34138717373460538339647393335, 7.60527673646873828448902392290, 8.121209127355723559965411973524, 9.306690668154280828992981774392, 10.38003321646047501128548409050, 11.15103890131745181688981374092, 12.60629648005311737832207163260

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