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Properties

Label 2-325-65.37-c1-0-9
Degree $2$
Conductor $325$
Sign $0.615 - 0.788i$
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  + (−1.80 + 1.04i)2-s + (2.66 + 0.713i)3-s + (1.17 − 2.04i)4-s + (−5.55 + 1.48i)6-s + (1.45 − 2.52i)7-s + 0.750i·8-s + (3.97 + 2.29i)9-s + (−0.0254 − 0.00681i)11-s + (4.59 − 4.59i)12-s + (3.56 + 0.530i)13-s + 6.07i·14-s + (1.57 + 2.72i)16-s + (0.741 + 2.76i)17-s − 9.58·18-s + (−1.23 − 4.62i)19-s + ⋯
L(s)  = 1  + (−1.27 + 0.738i)2-s + (1.53 + 0.411i)3-s + (0.589 − 1.02i)4-s + (−2.26 + 0.607i)6-s + (0.550 − 0.952i)7-s + 0.265i·8-s + (1.32 + 0.765i)9-s + (−0.00767 − 0.00205i)11-s + (1.32 − 1.32i)12-s + (0.989 + 0.147i)13-s + 1.62i·14-s + (0.393 + 0.682i)16-s + (0.179 + 0.671i)17-s − 2.26·18-s + (−0.284 − 1.06i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.615 - 0.788i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (232, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.615 - 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10570 + 0.539700i\)
\(L(\frac12)\) \(\approx\) \(1.10570 + 0.539700i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.56 - 0.530i)T \)
good2 \( 1 + (1.80 - 1.04i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-2.66 - 0.713i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.45 + 2.52i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.0254 + 0.00681i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.741 - 2.76i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.23 + 4.62i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.0961 + 0.358i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (3.62 - 2.09i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.835 + 0.835i)T + 31iT^{2} \)
37 \( 1 + (-3.22 - 5.58i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.02 - 7.57i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (6.69 - 1.79i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + 0.833T + 47T^{2} \)
53 \( 1 + (0.902 - 0.902i)T - 53iT^{2} \)
59 \( 1 + (1.44 - 0.387i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-5.35 + 9.26i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.6 - 6.15i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.57 - 0.957i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + 15.0iT - 73T^{2} \)
79 \( 1 + 4.25iT - 79T^{2} \)
83 \( 1 + 1.31T + 83T^{2} \)
89 \( 1 + (0.867 - 3.23i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.351 + 0.202i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.20371596637441092878325648809, −10.39434932156622324927831473247, −9.562622540081963626725984744544, −8.741027963635198631583480508156, −8.156026371360963968183262964668, −7.43913161313851848105998964025, −6.39725659866334289246091541106, −4.45940142066895881611064871445, −3.38288388221499765659032495202, −1.51903721127277850795326475061, 1.59627368833515543521388372912, 2.47643097513300988697432148987, 3.63575074173805769916917822214, 5.61885689998489277092793882513, 7.32756752436444374852258650190, 8.175405069948899127674054341200, 8.692577666423497692915603340789, 9.330707403070435134929149234112, 10.28843799920585494883547034415, 11.37302729541431502379827823064

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