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Properties

Label 2-325-13.4-c1-0-1
Degree $2$
Conductor $325$
Sign $0.177 - 0.984i$
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  + (−2.11 − 1.22i)2-s + (0.0609 − 0.105i)3-s + (1.98 + 3.44i)4-s + (−0.257 + 0.148i)6-s + (−0.358 + 0.207i)7-s − 4.82i·8-s + (1.49 + 2.58i)9-s + (−3.97 − 2.29i)11-s + 0.484·12-s + (−2.36 + 2.72i)13-s + 1.01·14-s + (−1.91 + 3.32i)16-s + (−0.372 − 0.644i)17-s − 7.29i·18-s + (−6.47 + 3.73i)19-s + ⋯
L(s)  = 1  + (−1.49 − 0.864i)2-s + (0.0351 − 0.0609i)3-s + (0.993 + 1.72i)4-s + (−0.105 + 0.0607i)6-s + (−0.135 + 0.0782i)7-s − 1.70i·8-s + (0.497 + 0.861i)9-s + (−1.19 − 0.691i)11-s + 0.139·12-s + (−0.655 + 0.755i)13-s + 0.270·14-s + (−0.479 + 0.831i)16-s + (−0.0902 − 0.156i)17-s − 1.71i·18-s + (−1.48 + 0.857i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.177 - 0.984i$
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: \((2,\ 325,\ (\ :1/2),\ 0.177 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.227436 + 0.190152i\)
\(L(\frac12)\) \(\approx\) \(0.227436 + 0.190152i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.36 - 2.72i)T \)
good2 \( 1 + (2.11 + 1.22i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.0609 + 0.105i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (0.358 - 0.207i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.97 + 2.29i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.372 + 0.644i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.47 - 3.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.17 - 5.50i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.80 + 4.85i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.60iT - 31T^{2} \)
37 \( 1 + (-5.57 - 3.21i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.51 - 2.60i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.96 - 8.60i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.52iT - 47T^{2} \)
53 \( 1 + 7.56T + 53T^{2} \)
59 \( 1 + (3.15 - 1.82i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.95 + 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.67 + 3.27i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.89 - 3.40i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.08iT - 73T^{2} \)
79 \( 1 - 1.13T + 79T^{2} \)
83 \( 1 + 12.1iT - 83T^{2} \)
89 \( 1 + (-6.04 - 3.48i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.0 + 6.35i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.44802238310324300322688319655, −10.69783884846270544521046198815, −10.02971741095443815712253270785, −9.201644720661352503319775185511, −7.940376027246813789914118343412, −7.78319874416878656609112645102, −6.20604567488643145627661739144, −4.55731415482343977295801542818, −2.85803243941989938867647444984, −1.81356005966812206415007407291, 0.32154789985711386939701456160, 2.38538453759267210300621021716, 4.50087740828243849392083044096, 5.92343468890952929413705692336, 6.87708398928204914620492387348, 7.63655594159426590246184690148, 8.556346246814333452580341713892, 9.427435827710622024393577146147, 10.32297164993288160948684298045, 10.71979644048979074356072472748

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