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Properties

Label 2-245-245.103-c1-0-14
Degree $2$
Conductor $245$
Sign $0.865 - 0.500i$
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  + (−0.0698 + 0.0946i)2-s + (2.41 − 0.456i)3-s + (0.585 + 1.89i)4-s + (−1.81 + 1.29i)5-s + (−0.125 + 0.260i)6-s + (1.69 − 2.03i)7-s + (−0.442 − 0.154i)8-s + (2.82 − 1.10i)9-s + (0.00412 − 0.263i)10-s + (−0.120 + 0.305i)11-s + (2.28 + 4.31i)12-s + (1.21 + 0.137i)13-s + (0.0737 + 0.302i)14-s + (−3.79 + 3.96i)15-s + (−3.23 + 2.20i)16-s + (0.851 + 0.0318i)17-s + ⋯
L(s)  = 1  + (−0.0494 + 0.0669i)2-s + (1.39 − 0.263i)3-s + (0.292 + 0.948i)4-s + (−0.813 + 0.581i)5-s + (−0.0512 + 0.106i)6-s + (0.640 − 0.767i)7-s + (−0.156 − 0.0547i)8-s + (0.942 − 0.369i)9-s + (0.00130 − 0.0831i)10-s + (−0.0361 + 0.0922i)11-s + (0.658 + 1.24i)12-s + (0.337 + 0.0380i)13-s + (0.0197 + 0.0808i)14-s + (−0.981 + 1.02i)15-s + (−0.809 + 0.551i)16-s + (0.206 + 0.00773i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.865 - 0.500i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 0.865 - 0.500i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71676 + 0.460264i\)
\(L(\frac12)\) \(\approx\) \(1.71676 + 0.460264i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.81 - 1.29i)T \)
7 \( 1 + (-1.69 + 2.03i)T \)
good2 \( 1 + (0.0698 - 0.0946i)T + (-0.589 - 1.91i)T^{2} \)
3 \( 1 + (-2.41 + 0.456i)T + (2.79 - 1.09i)T^{2} \)
11 \( 1 + (0.120 - 0.305i)T + (-8.06 - 7.48i)T^{2} \)
13 \( 1 + (-1.21 - 0.137i)T + (12.6 + 2.89i)T^{2} \)
17 \( 1 + (-0.851 - 0.0318i)T + (16.9 + 1.27i)T^{2} \)
19 \( 1 + (1.69 - 2.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.287 + 7.67i)T + (-22.9 + 1.71i)T^{2} \)
29 \( 1 + (8.41 - 1.92i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-8.13 + 4.69i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.69 - 1.42i)T + (20.8 - 30.5i)T^{2} \)
41 \( 1 + (3.82 + 7.93i)T + (-25.5 + 32.0i)T^{2} \)
43 \( 1 + (-0.640 - 1.83i)T + (-33.6 + 26.8i)T^{2} \)
47 \( 1 + (-3.54 - 2.61i)T + (13.8 + 44.9i)T^{2} \)
53 \( 1 + (4.73 + 2.50i)T + (29.8 + 43.7i)T^{2} \)
59 \( 1 + (-0.633 - 8.45i)T + (-58.3 + 8.79i)T^{2} \)
61 \( 1 + (-2.31 + 7.48i)T + (-50.4 - 34.3i)T^{2} \)
67 \( 1 + (-1.73 - 0.465i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.413 + 1.81i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (4.30 - 3.17i)T + (21.5 - 69.7i)T^{2} \)
79 \( 1 + (6.73 + 3.88i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.40 - 12.4i)T + (-80.9 + 18.4i)T^{2} \)
89 \( 1 + (0.319 + 0.813i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-10.5 - 10.5i)T + 97iT^{2} \)
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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

−12.24994780042960179104167961404, −11.26368816842480631389669857685, −10.33531437068492537242624156246, −8.802414582210781271080934441933, −8.081513317484871648015406053586, −7.58319285641354755555142305022, −6.65320384260970198183746513968, −4.23388391847058664889562716552, −3.50272208948586900676121401810, −2.29519143837248401237419283648, 1.73576942873223434955064920236, 3.18680763071696161993581195002, 4.58876908895335317122637210429, 5.69825606746902540400699513711, 7.35361044064832615348136117732, 8.347460931225057682157438508021, 9.001351574925219031162145328839, 9.820137370590556414725231262937, 11.17413262940080206726006120697, 11.79597095819413741591049647729

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