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Properties

Label 2-1925-385.153-c0-0-7
Degree $2$
Conductor $1925$
Sign $-0.608 - 0.793i$
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  + (−0.707 − 0.707i)2-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + i·9-s + (−0.5 − 0.866i)11-s + 1.00i·14-s + 1.00·16-s + (0.707 − 0.707i)18-s + (−0.258 + 0.965i)22-s + (−1.22 − 1.22i)23-s − 1.73·29-s + (−1.22 + 1.22i)37-s + (−0.707 + 0.707i)43-s + 1.73i·46-s + 1.00i·49-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + i·9-s + (−0.5 − 0.866i)11-s + 1.00i·14-s + 1.00·16-s + (0.707 − 0.707i)18-s + (−0.258 + 0.965i)22-s + (−1.22 − 1.22i)23-s − 1.73·29-s + (−1.22 + 1.22i)37-s + (−0.707 + 0.707i)43-s + 1.73i·46-s + 1.00i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $-0.608 - 0.793i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1693, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ -0.608 - 0.793i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1194684255\)
\(L(\frac12)\) \(\approx\) \(0.1194684255\)
\(L(1)\) 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 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
3 \( 1 - iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
29 \( 1 + 1.73T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + 1.73T + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + iT^{2} \)
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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

−8.937413160832672828205657195167, −8.270163754974547660122480985402, −7.58314519156258640593756672012, −6.47242048375655178568754236577, −5.72419664130496760936268464042, −4.81206923411779520275639380938, −3.60224581623420799670860079948, −2.68764483924140092030201014520, −1.66675912390277921664476638977, −0.10323774396426775483792737637, 2.01228577952821758886545043971, 3.31675212690511534777902824405, 3.94295148761655084680918762609, 5.51034543517896651496282733433, 5.99352665611368500728085890719, 7.04924179633465435539268661114, 7.37068423544754272685901805327, 8.426281497757717237863479170911, 9.067846197837611036118047650922, 9.685186851915361262946685778992

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