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Properties

Label 2-425-5.4-c3-0-29
Degree $2$
Conductor $425$
Sign $0.894 - 0.447i$
Analytic cond. $25.0758$
Root an. cond. $5.00757$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.36i·2-s + 3.15i·3-s + 6.14·4-s + 4.29·6-s + 7.94i·7-s − 19.2i·8-s + 17.0·9-s + 27.6·11-s + 19.3i·12-s + 58.1i·13-s + 10.8·14-s + 22.9·16-s + 17i·17-s − 23.2i·18-s − 89.1·19-s + ⋯
L(s)  = 1  − 0.481i·2-s + 0.607i·3-s + 0.768·4-s + 0.292·6-s + 0.428i·7-s − 0.851i·8-s + 0.631·9-s + 0.756·11-s + 0.466i·12-s + 1.23i·13-s + 0.206·14-s + 0.358·16-s + 0.242i·17-s − 0.303i·18-s − 1.07·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(25.0758\)
Root analytic conductor: \(5.00757\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :3/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.636418536\)
\(L(\frac12)\) \(\approx\) \(2.636418536\)
\(L(\frac{5}{2})\) 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 \( 1 - 17iT \)
good2 \( 1 + 1.36iT - 8T^{2} \)
3 \( 1 - 3.15iT - 27T^{2} \)
7 \( 1 - 7.94iT - 343T^{2} \)
11 \( 1 - 27.6T + 1.33e3T^{2} \)
13 \( 1 - 58.1iT - 2.19e3T^{2} \)
19 \( 1 + 89.1T + 6.85e3T^{2} \)
23 \( 1 + 115. iT - 1.21e4T^{2} \)
29 \( 1 - 128.T + 2.43e4T^{2} \)
31 \( 1 - 273.T + 2.97e4T^{2} \)
37 \( 1 - 132. iT - 5.06e4T^{2} \)
41 \( 1 + 470.T + 6.89e4T^{2} \)
43 \( 1 - 352. iT - 7.95e4T^{2} \)
47 \( 1 + 152. iT - 1.03e5T^{2} \)
53 \( 1 - 527. iT - 1.48e5T^{2} \)
59 \( 1 - 292.T + 2.05e5T^{2} \)
61 \( 1 + 53.8T + 2.26e5T^{2} \)
67 \( 1 + 52.9iT - 3.00e5T^{2} \)
71 \( 1 - 788.T + 3.57e5T^{2} \)
73 \( 1 - 295. iT - 3.89e5T^{2} \)
79 \( 1 - 720.T + 4.93e5T^{2} \)
83 \( 1 + 116. iT - 5.71e5T^{2} \)
89 \( 1 - 813.T + 7.04e5T^{2} \)
97 \( 1 + 794. iT - 9.12e5T^{2} \)
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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

−10.72308870657567844805290598227, −10.11629982777248473491691629807, −9.223626366010958020661931753624, −8.261910977690459780226048112589, −6.71165125949716709475273442944, −6.44961528817316771571179940816, −4.67928101884117464611030283685, −3.90021532882361645014123673841, −2.52850687783154344017929280299, −1.37657490741209906734996250300, 0.960096608816025830081263783952, 2.21727334428054659638299200789, 3.64420917737192377282506333751, 5.09055560433050006661600839449, 6.31443601514253187913459317505, 6.88966437912808954947995390043, 7.75663756959204948465481190503, 8.502704083337226475699960330998, 9.986759146171986558493080384504, 10.62749750277293710057057871285

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