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Properties

Label 2-325-13.9-c1-0-11
Degree $2$
Conductor $325$
Sign $0.962 - 0.271i$
Analytic cond. $2.59513$
Root an. cond. $1.61094$
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  + (−1.27 + 2.20i)2-s + (−1.07 + 1.86i)3-s + (−2.24 − 3.88i)4-s + (−2.74 − 4.74i)6-s + (−1.46 − 2.54i)7-s + 6.31·8-s + (−0.817 − 1.41i)9-s + (0.317 − 0.550i)11-s + 9.64·12-s + (−0.0716 − 3.60i)13-s + 7.48·14-s + (−3.55 + 6.16i)16-s + (−0.611 − 1.05i)17-s + 4.16·18-s + (−0.682 − 1.18i)19-s + ⋯
L(s)  = 1  + (−0.900 + 1.55i)2-s + (−0.621 + 1.07i)3-s + (−1.12 − 1.94i)4-s + (−1.11 − 1.93i)6-s + (−0.555 − 0.961i)7-s + 2.23·8-s + (−0.272 − 0.472i)9-s + (0.0957 − 0.165i)11-s + 2.78·12-s + (−0.0198 − 0.999i)13-s + 1.99·14-s + (−0.889 + 1.54i)16-s + (−0.148 − 0.257i)17-s + 0.981·18-s + (−0.156 − 0.271i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.962 - 0.271i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.962 - 0.271i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.321511 + 0.0445323i\)
\(L(\frac12)\) \(\approx\) \(0.321511 + 0.0445323i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 + (0.0716 + 3.60i)T \)
good2 \( 1 + (1.27 - 2.20i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.07 - 1.86i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (1.46 + 2.54i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.317 + 0.550i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.611 + 1.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.682 + 1.18i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.07 + 1.86i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.96T + 31T^{2} \)
37 \( 1 + (-0.611 + 1.05i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.98 + 8.62i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.683 + 1.18i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.16T + 47T^{2} \)
53 \( 1 + 0.642T + 53T^{2} \)
59 \( 1 + (3.79 + 6.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.13 - 1.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.01 + 6.95i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.31 + 2.28i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 10.3T + 73T^{2} \)
79 \( 1 - 1.03T + 79T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 + (-6.27 + 10.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.39 + 12.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.92170075198947535681153754756, −10.52584730786406572976888174895, −9.644446408482332870868481937290, −8.921837887794307421561953639868, −7.67851550299640979567448072412, −6.93624391417427160253158638101, −5.80924403198347602223039898062, −5.05285195064736343337465957630, −3.82628270283402294002025052495, −0.35153418649211656866614430062, 1.47798420532240754271086828667, 2.51296369904954113382691381509, 3.98468387414823260351050599000, 5.78891947910469262592889551719, 6.90810412599483757226532316061, 8.009285759574108533886892975983, 9.145783057831013364512599649215, 9.607954370289128789237173991285, 10.94939794294647601446255928946, 11.57498532207817435327416615293

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