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Properties

Label 2-325-13.4-c1-0-13
Degree $2$
Conductor $325$
Sign $0.252 + 0.967i$
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  + (0.395 + 0.228i)2-s + (0.5 − 0.866i)3-s + (−0.895 − 1.55i)4-s + (0.395 − 0.228i)6-s + (1.5 − 0.866i)7-s − 1.73i·8-s + (1 + 1.73i)9-s + (−2.29 − 1.32i)11-s − 1.79·12-s + (1 − 3.46i)13-s + 0.791·14-s + (−1.39 + 2.41i)16-s + (−2.29 − 3.96i)17-s + 0.913i·18-s + (1.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (0.279 + 0.161i)2-s + (0.288 − 0.499i)3-s + (−0.447 − 0.775i)4-s + (0.161 − 0.0932i)6-s + (0.566 − 0.327i)7-s − 0.612i·8-s + (0.333 + 0.577i)9-s + (−0.690 − 0.398i)11-s − 0.517·12-s + (0.277 − 0.960i)13-s + 0.211·14-s + (−0.348 + 0.604i)16-s + (−0.555 − 0.962i)17-s + 0.215i·18-s + (0.344 − 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.252 + 0.967i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.252 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20469 - 0.930548i\)
\(L(\frac12)\) \(\approx\) \(1.20469 - 0.930548i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 3.46i)T \)
good2 \( 1 + (-0.395 - 0.228i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-1.5 + 0.866i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.29 + 1.32i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.29 + 3.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.5 + 0.866i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.29 + 3.96i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.29 - 3.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.20iT - 31T^{2} \)
37 \( 1 + (-6.87 - 3.96i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.29 - 1.32i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.29 - 9.16i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.82iT - 47T^{2} \)
53 \( 1 + 7.58T + 53T^{2} \)
59 \( 1 + (-12.0 + 6.97i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.708 - 1.22i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.873 + 0.504i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.08 + 3.51i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 - 6.01iT - 83T^{2} \)
89 \( 1 + (8.29 + 4.78i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.87 + 5.70i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.12643783273758568370648052400, −10.64900111540290268594837361452, −9.575109765534969514596512098443, −8.439062313850353571922820828390, −7.61688080187266661322958541001, −6.55684638700484600407611461967, −5.26231182939316921901199899126, −4.61881381391856813205403898449, −2.82305398652931461650624333145, −1.08853663304724741854885715327, 2.25491965541077868038895123401, 3.78109682012003773829456612159, 4.41246719101620541569166528726, 5.67344096899098389363970767558, 7.17407283803751935450407346878, 8.157835554837272397313791968929, 9.031617414698767259253966033692, 9.753441988032109719637690104154, 11.07025633958003997010517357092, 11.80705968633049390915754154509

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