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Properties

Label 2-245-35.17-c1-0-13
Degree $2$
Conductor $245$
Sign $-0.578 + 0.815i$
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  + (0.5 − 1.86i)2-s + (1.86 − 0.5i)3-s + (−1.5 − 0.866i)4-s + (−1.86 − 1.23i)5-s − 3.73i·6-s + (0.366 − 0.366i)8-s + (0.633 − 0.366i)9-s + (−3.23 + 2.86i)10-s + (−0.366 + 0.633i)11-s + (−3.23 − 0.866i)12-s + (2 + 2i)13-s + (−4.09 − 1.36i)15-s + (−2.23 − 3.86i)16-s + (−0.267 − i)17-s + (−0.366 − 1.36i)18-s + (1.36 + 2.36i)19-s + ⋯
L(s)  = 1  + (0.353 − 1.31i)2-s + (1.07 − 0.288i)3-s + (−0.750 − 0.433i)4-s + (−0.834 − 0.550i)5-s − 1.52i·6-s + (0.129 − 0.129i)8-s + (0.211 − 0.122i)9-s + (−1.02 + 0.906i)10-s + (−0.110 + 0.191i)11-s + (−0.933 − 0.249i)12-s + (0.554 + 0.554i)13-s + (−1.05 − 0.352i)15-s + (−0.558 − 0.966i)16-s + (−0.0649 − 0.242i)17-s + (−0.0862 − 0.321i)18-s + (0.313 + 0.542i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.578 + 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.578 + 0.815i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ -0.578 + 0.815i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.840842 - 1.62731i\)
\(L(\frac12)\) \(\approx\) \(0.840842 - 1.62731i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.86 + 1.23i)T \)
7 \( 1 \)
good2 \( 1 + (-0.5 + 1.86i)T + (-1.73 - i)T^{2} \)
3 \( 1 + (-1.86 + 0.5i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (0.366 - 0.633i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2 - 2i)T + 13iT^{2} \)
17 \( 1 + (0.267 + i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.36 - 2.36i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.96 - 1.86i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 + (0.464 + 0.267i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.26 - 4.73i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.464iT - 41T^{2} \)
43 \( 1 + (5.83 - 5.83i)T - 43iT^{2} \)
47 \( 1 + (0.633 + 0.169i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.83 + 6.83i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.09 - 1.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.33 + 4.23i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.13 + 0.303i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 4.73T + 71T^{2} \)
73 \( 1 + (3.46 - 0.928i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (5.83 - 3.36i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.09 + 3.09i)T + 83iT^{2} \)
89 \( 1 + (8.33 + 14.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.92 - 7.92i)T - 97iT^{2} \)
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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

−11.66310594012435834808404894845, −11.21163984511522956499602054964, −9.848432597089030718797897489354, −8.949612875179780257974502966109, −8.050188143434035454381855578629, −7.03993690360334365223879399408, −4.99081220727693525348070557681, −3.81980212625441997279938671993, −2.95724048426322936430357338095, −1.50465327764141034868168190830, 2.90988571243839721849702450456, 3.98184147530160152114854506620, 5.30417541739839686574351533458, 6.62611938972739241586156932095, 7.47445748740536235799365581415, 8.352470731670388017030451170634, 8.969684289294999022759654612057, 10.53576436084373609933076237497, 11.35600304087761247675586432425, 12.82867069991793461784172923522

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