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Properties

Label 2-425-17.15-c1-0-10
Degree $2$
Conductor $425$
Sign $0.992 - 0.125i$
Analytic cond. $3.39364$
Root an. cond. $1.84218$
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  + (0.176 + 0.176i)2-s + (0.629 + 1.51i)3-s − 1.93i·4-s + (−0.156 + 0.377i)6-s + (1.32 + 0.547i)7-s + (0.693 − 0.693i)8-s + (0.210 − 0.210i)9-s + (1.03 − 2.48i)11-s + (2.94 − 1.21i)12-s − 0.174i·13-s + (0.136 + 0.329i)14-s − 3.63·16-s + (3.44 + 2.27i)17-s + 0.0742·18-s + (2.69 + 2.69i)19-s + ⋯
L(s)  = 1  + (0.124 + 0.124i)2-s + (0.363 + 0.876i)3-s − 0.969i·4-s + (−0.0639 + 0.154i)6-s + (0.499 + 0.206i)7-s + (0.245 − 0.245i)8-s + (0.0703 − 0.0703i)9-s + (0.310 − 0.750i)11-s + (0.849 − 0.351i)12-s − 0.0484i·13-s + (0.0364 + 0.0879i)14-s − 0.908·16-s + (0.834 + 0.550i)17-s + 0.0175·18-s + (0.618 + 0.618i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.992 - 0.125i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.992 - 0.125i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81878 + 0.114672i\)
\(L(\frac12)\) \(\approx\) \(1.81878 + 0.114672i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 + (-3.44 - 2.27i)T \)
good2 \( 1 + (-0.176 - 0.176i)T + 2iT^{2} \)
3 \( 1 + (-0.629 - 1.51i)T + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (-1.32 - 0.547i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.03 + 2.48i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 0.174iT - 13T^{2} \)
19 \( 1 + (-2.69 - 2.69i)T + 19iT^{2} \)
23 \( 1 + (-1.05 + 2.54i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (5.94 - 2.46i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (0.188 + 0.454i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (2.19 + 5.30i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-5.54 - 2.29i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (8.00 - 8.00i)T - 43iT^{2} \)
47 \( 1 - 2.65iT - 47T^{2} \)
53 \( 1 + (-8.73 - 8.73i)T + 53iT^{2} \)
59 \( 1 + (-5.72 + 5.72i)T - 59iT^{2} \)
61 \( 1 + (6.41 + 2.65i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 12.3T + 67T^{2} \)
71 \( 1 + (4.67 + 11.2i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (14.4 - 5.99i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-4.94 + 11.9i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (6.06 + 6.06i)T + 83iT^{2} \)
89 \( 1 - 11.4iT - 89T^{2} \)
97 \( 1 + (-2.12 + 0.879i)T + (68.5 - 68.5i)T^{2} \)
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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

−10.93657615581148394663115873542, −10.27598680256629495008049361262, −9.445228710601528395572300914253, −8.735106354963821777928523411110, −7.55080499350110094808107154516, −6.21444092878420178142492490636, −5.41479460383212344864650882192, −4.37938545996250424354046141744, −3.29202967985801944558037125588, −1.41865797100854894705232483452, 1.63737588374589903280645595083, 2.86840103022583479557647094527, 4.14630158024493166988754554113, 5.26810366157181470362370925025, 7.04357706677537992517050631747, 7.34213141648999907197423073257, 8.195362966036093791769829298543, 9.169804992868656060339630923095, 10.27807193712079591498015578441, 11.62486775964501421983267319769

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