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Properties

Label 2-1925-385.13-c0-0-2
Degree $2$
Conductor $1925$
Sign $0.304 - 0.952i$
Analytic cond. $0.960700$
Root an. cond. $0.980153$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.186 + 0.0949i)2-s + (−0.562 + 0.773i)4-s + (0.156 + 0.987i)7-s + (0.0639 − 0.403i)8-s + (0.951 + 0.309i)9-s + (0.669 − 0.743i)11-s + (−0.122 − 0.169i)14-s + (−0.269 − 0.828i)16-s + (−0.206 + 0.0327i)18-s + (−0.0541 + 0.201i)22-s + (0.575 + 0.575i)23-s + (−0.852 − 0.434i)28-s + (1.20 + 0.873i)29-s + (0.417 + 0.417i)32-s + (−0.773 + 0.562i)36-s + (−1.96 + 0.311i)37-s + ⋯
L(s)  = 1  + (−0.186 + 0.0949i)2-s + (−0.562 + 0.773i)4-s + (0.156 + 0.987i)7-s + (0.0639 − 0.403i)8-s + (0.951 + 0.309i)9-s + (0.669 − 0.743i)11-s + (−0.122 − 0.169i)14-s + (−0.269 − 0.828i)16-s + (−0.206 + 0.0327i)18-s + (−0.0541 + 0.201i)22-s + (0.575 + 0.575i)23-s + (−0.852 − 0.434i)28-s + (1.20 + 0.873i)29-s + (0.417 + 0.417i)32-s + (−0.773 + 0.562i)36-s + (−1.96 + 0.311i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.304 - 0.952i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ 0.304 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.024405822\)
\(L(\frac12)\) \(\approx\) \(1.024405822\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + (-0.156 - 0.987i)T \)
11 \( 1 + (-0.669 + 0.743i)T \)
good2 \( 1 + (0.186 - 0.0949i)T + (0.587 - 0.809i)T^{2} \)
3 \( 1 + (-0.951 - 0.309i)T^{2} \)
13 \( 1 + (0.587 - 0.809i)T^{2} \)
17 \( 1 + (0.587 + 0.809i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.575 - 0.575i)T + iT^{2} \)
29 \( 1 + (-1.20 - 0.873i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (1.96 - 0.311i)T + (0.951 - 0.309i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 + (-0.946 + 0.946i)T - iT^{2} \)
47 \( 1 + (0.951 + 0.309i)T^{2} \)
53 \( 1 + (1.69 - 0.863i)T + (0.587 - 0.809i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.294 - 0.294i)T - iT^{2} \)
71 \( 1 + (-0.413 - 1.27i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.951 + 0.309i)T^{2} \)
79 \( 1 + (-0.128 + 0.395i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.587 - 0.809i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.587 - 0.809i)T^{2} \)
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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

−9.243124991013268582640054308742, −8.820403816571823110844816387344, −8.139172205907412451976871130778, −7.25142314708506936356949530575, −6.55082605439163696133845523532, −5.41355733213686720258519316223, −4.68306077799460251923912322846, −3.70058092324462362260604402800, −2.87080416112415001176197864034, −1.44962686677197694790050365163, 0.961300027798154696964468146978, 1.89315208989358388923526471343, 3.57288158098420703792407441264, 4.50529128008342251736136627302, 4.84364038511508585360310878600, 6.24721636323904254343474025950, 6.81383026971355291397654921871, 7.63374758998664778661762594238, 8.582027685559480650945788589221, 9.440595889666888589451326834030

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