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Properties

Label 2-325-65.58-c1-0-4
Degree $2$
Conductor $325$
Sign $0.998 + 0.0463i$
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  + (−0.791 − 0.457i)2-s + (0.325 − 0.0872i)3-s + (−0.582 − 1.00i)4-s + (−0.297 − 0.0797i)6-s + (2.08 + 3.61i)7-s + 2.89i·8-s + (−2.49 + 1.44i)9-s + (5.01 − 1.34i)11-s + (−0.277 − 0.277i)12-s + (2.17 + 2.87i)13-s − 3.81i·14-s + (0.157 − 0.272i)16-s + (0.780 − 2.91i)17-s + 2.63·18-s + (1.43 − 5.35i)19-s + ⋯
L(s)  = 1  + (−0.559 − 0.323i)2-s + (0.188 − 0.0503i)3-s + (−0.291 − 0.504i)4-s + (−0.121 − 0.0325i)6-s + (0.788 + 1.36i)7-s + 1.02i·8-s + (−0.833 + 0.481i)9-s + (1.51 − 0.404i)11-s + (−0.0801 − 0.0801i)12-s + (0.602 + 0.798i)13-s − 1.01i·14-s + (0.0393 − 0.0682i)16-s + (0.189 − 0.706i)17-s + 0.621·18-s + (0.329 − 1.22i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0463i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.998 + 0.0463i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (318, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.998 + 0.0463i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06529 - 0.0247080i\)
\(L(\frac12)\) \(\approx\) \(1.06529 - 0.0247080i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 - 2.87i)T \)
good2 \( 1 + (0.791 + 0.457i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.325 + 0.0872i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-2.08 - 3.61i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.01 + 1.34i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.780 + 2.91i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.43 + 5.35i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.442 + 1.65i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-7.08 - 4.08i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.64 - 2.64i)T - 31iT^{2} \)
37 \( 1 + (3.88 - 6.72i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.228 + 0.853i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.839 - 0.224i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 1.44T + 47T^{2} \)
53 \( 1 + (-0.405 - 0.405i)T + 53iT^{2} \)
59 \( 1 + (-9.44 - 2.53i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.28 + 3.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.15 + 3.55i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.65 + 0.712i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 6.02iT - 73T^{2} \)
79 \( 1 - 12.5iT - 79T^{2} \)
83 \( 1 + 8.44T + 83T^{2} \)
89 \( 1 + (3.79 + 14.1i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.670 + 0.386i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.55886737708052672935443177422, −10.85037052211601895535102125491, −9.410593831506701184165872858818, −8.774137840971385682424182896686, −8.458675395514095771517656621889, −6.69873333965240131432879055221, −5.58453920760557684297571515158, −4.72347056315860818698794998371, −2.79502535163952698028313831834, −1.49594678648203931407420085066, 1.12088687216441820258015852153, 3.62179119740537369063834009582, 4.12466238697601262201109222095, 5.93200790572687251379725938132, 7.06072582045606025680774924819, 7.980315767259822651825549247659, 8.589932930786150539745432278600, 9.652093623189481029843662520046, 10.50597857364789196458799776446, 11.63368736690484327182474468564

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