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Properties

Label 2-1925-385.167-c0-0-0
Degree $2$
Conductor $1925$
Sign $-0.247 + 0.968i$
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  + (0.829 + 1.62i)2-s + (−1.37 + 1.89i)4-s + (−0.987 + 0.156i)7-s + (−2.41 − 0.382i)8-s + (−0.951 − 0.309i)9-s + (−0.978 − 0.207i)11-s + (−1.07 − 1.47i)14-s + (−0.657 − 2.02i)16-s + (−0.285 − 1.80i)18-s + (−0.472 − 1.76i)22-s + (−1.40 + 1.40i)23-s + (1.06 − 2.08i)28-s + (−0.336 − 0.244i)29-s + (1.02 − 1.02i)32-s + (1.89 − 1.37i)36-s + (0.127 + 0.803i)37-s + ⋯
L(s)  = 1  + (0.829 + 1.62i)2-s + (−1.37 + 1.89i)4-s + (−0.987 + 0.156i)7-s + (−2.41 − 0.382i)8-s + (−0.951 − 0.309i)9-s + (−0.978 − 0.207i)11-s + (−1.07 − 1.47i)14-s + (−0.657 − 2.02i)16-s + (−0.285 − 1.80i)18-s + (−0.472 − 1.76i)22-s + (−1.40 + 1.40i)23-s + (1.06 − 2.08i)28-s + (−0.336 − 0.244i)29-s + (1.02 − 1.02i)32-s + (1.89 − 1.37i)36-s + (0.127 + 0.803i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $-0.247 + 0.968i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ -0.247 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6218158984\)
\(L(\frac12)\) \(\approx\) \(0.6218158984\)
\(L(1)\) 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 \( 1 + (0.987 - 0.156i)T \)
11 \( 1 + (0.978 + 0.207i)T \)
good2 \( 1 + (-0.829 - 1.62i)T + (-0.587 + 0.809i)T^{2} \)
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 \( 1 + (1.40 - 1.40i)T - iT^{2} \)
29 \( 1 + (0.336 + 0.244i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.127 - 0.803i)T + (-0.951 + 0.309i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 + (-1.38 - 1.38i)T + iT^{2} \)
47 \( 1 + (-0.951 - 0.309i)T^{2} \)
53 \( 1 + (-0.863 - 1.69i)T + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (1.05 + 1.05i)T + iT^{2} \)
71 \( 1 + (0.604 + 1.86i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.951 - 0.309i)T^{2} \)
79 \( 1 + (0.459 - 1.41i)T + (-0.809 - 0.587i)T^{2} \)
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

−9.527144941236414824993765156592, −8.978238901920901878082555756599, −7.960871850978633462514258172725, −7.63816104161360967032189261511, −6.55855865010884197023085266426, −5.85320793853318961418145454359, −5.58884612498809677750671684902, −4.42562848948401655920145847993, −3.49314311581602154351946930565, −2.75085424498444071256928499911, 0.31641803019939587858966947539, 2.20630641466667113065843448926, 2.70075000534738892887799412826, 3.67625398937166860032486512987, 4.43554174618006391040991544093, 5.51968007958904939361202595941, 5.92744657961785263542271465773, 7.18121472252996837875605425477, 8.374905645989230804684190354059, 9.098490805633598148768696614725

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