Properties

Label 2-1925-385.153-c0-0-4
Degree $2$
Conductor $1925$
Sign $0.382 - 0.923i$
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 + (−0.965 + 0.258i)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 + (−0.965 + 0.258i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.382 - 0.923i$
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: \((2,\ 1925,\ (\ :0),\ 0.382 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.801632878\)
\(L(\frac12)\) \(\approx\) \(1.801632878\)
\(L(1)\) 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

−9.602425835678759839376886930620, −8.347227561603601960353654643875, −8.015168285218385262371529013193, −7.03058185419447384143089564355, −6.30424903774231907587394106573, −5.34756074085885972361469411069, −4.86115429932486672353366002324, −4.22162507074858748592737711805, −2.62412005679996567531933697609, −1.73202468792968190109525670572, 1.21639681428285370711287025162, 2.49334557247972217385904514221, 3.54730265318532926737536231250, 4.02201433090993204038494427099, 5.03489902729095726799909507247, 5.82445932643668838864506844204, 6.88277911002229441516377273589, 7.82629769864416755633766877862, 8.313775977605543410275583287431, 9.293651425201726605739648226486

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