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Properties

Label 2-325-5.4-c5-0-19
Degree $2$
Conductor $325$
Sign $-0.447 - 0.894i$
Analytic cond. $52.1247$
Root an. cond. $7.21974$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.1i·2-s − 24.5i·3-s − 70.0·4-s + 247.·6-s + 72.1i·7-s − 384. i·8-s − 358.·9-s + 127.·11-s + 1.71e3i·12-s + 169i·13-s − 728.·14-s + 1.64e3·16-s − 2.15e3i·17-s − 3.61e3i·18-s − 2.72e3·19-s + ⋯
L(s)  = 1  + 1.78i·2-s − 1.57i·3-s − 2.18·4-s + 2.80·6-s + 0.556i·7-s − 2.12i·8-s − 1.47·9-s + 0.317·11-s + 3.44i·12-s + 0.277i·13-s − 0.993·14-s + 1.60·16-s − 1.80i·17-s − 2.63i·18-s − 1.73·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(52.1247\)
Root analytic conductor: \(7.21974\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :5/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.226203782\)
\(L(\frac12)\) \(\approx\) \(1.226203782\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 - 169iT \)
good2 \( 1 - 10.1iT - 32T^{2} \)
3 \( 1 + 24.5iT - 243T^{2} \)
7 \( 1 - 72.1iT - 1.68e4T^{2} \)
11 \( 1 - 127.T + 1.61e5T^{2} \)
17 \( 1 + 2.15e3iT - 1.41e6T^{2} \)
19 \( 1 + 2.72e3T + 2.47e6T^{2} \)
23 \( 1 - 2.65e3iT - 6.43e6T^{2} \)
29 \( 1 - 5.88e3T + 2.05e7T^{2} \)
31 \( 1 - 1.36e3T + 2.86e7T^{2} \)
37 \( 1 - 481. iT - 6.93e7T^{2} \)
41 \( 1 - 7.38e3T + 1.15e8T^{2} \)
43 \( 1 - 1.01e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.63e4iT - 2.29e8T^{2} \)
53 \( 1 + 9.06e3iT - 4.18e8T^{2} \)
59 \( 1 + 2.95e4T + 7.14e8T^{2} \)
61 \( 1 - 1.64e4T + 8.44e8T^{2} \)
67 \( 1 - 6.15e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.22e3T + 1.80e9T^{2} \)
73 \( 1 - 6.78e4iT - 2.07e9T^{2} \)
79 \( 1 - 8.95e4T + 3.07e9T^{2} \)
83 \( 1 - 7.89e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.11e5T + 5.58e9T^{2} \)
97 \( 1 - 7.15e4iT - 8.58e9T^{2} \)
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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

−11.39866621158612150537494877409, −9.554444891715790156695952603413, −8.692238740307307371459756765837, −7.949067440612872588068863616525, −7.07262865687862113074382046392, −6.50722936244026230425058337375, −5.67984795001947724957616893709, −4.51924846111782127497741709882, −2.50781147978392259357850902122, −0.875135445223271934174466190226, 0.43031295940129504376659832031, 2.02993396173403012153945068077, 3.34818940643426416303641401175, 4.16259405974190355586257935713, 4.65205638026373899382546098118, 6.20197205332137580795549425688, 8.397743745190988775626943034507, 8.935077243303355664054559841755, 10.09778378512772204987001551623, 10.57127637115679268052131421827

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