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Properties

Label 2-1525-305.304-c1-0-85
Degree $2$
Conductor $1525$
Sign $0.0644 + 0.997i$
Analytic cond. $12.1771$
Root an. cond. $3.48958$
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  + 2.39·2-s − 2.73i·3-s + 3.73·4-s − 6.54i·6-s + 4.14·7-s + 4.14·8-s − 4.46·9-s − 2.39i·11-s − 10.1i·12-s + 3i·13-s + 9.92·14-s + 2.46·16-s + 1.75·17-s − 10.6·18-s − 4.73·19-s + ⋯
L(s)  = 1  + 1.69·2-s − 1.57i·3-s + 1.86·4-s − 2.67i·6-s + 1.56·7-s + 1.46·8-s − 1.48·9-s − 0.721i·11-s − 2.94i·12-s + 0.832i·13-s + 2.65·14-s + 0.616·16-s + 0.425·17-s − 2.51·18-s − 1.08·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0644 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0644 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1525\)    =    \(5^{2} \cdot 61\)
Sign: $0.0644 + 0.997i$
Analytic conductor: \(12.1771\)
Root analytic conductor: \(3.48958\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1525} (1524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1525,\ (\ :1/2),\ 0.0644 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(5.184499883\)
\(L(\frac12)\) \(\approx\) \(5.184499883\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
61 \( 1 + (-7.19 - 3.03i)T \)
good2 \( 1 - 2.39T + 2T^{2} \)
3 \( 1 + 2.73iT - 3T^{2} \)
7 \( 1 - 4.14T + 7T^{2} \)
11 \( 1 + 2.39iT - 11T^{2} \)
13 \( 1 - 3iT - 13T^{2} \)
17 \( 1 - 1.75T + 17T^{2} \)
19 \( 1 + 4.73T + 19T^{2} \)
23 \( 1 - 5.89T + 23T^{2} \)
29 \( 1 - 1.75iT - 29T^{2} \)
31 \( 1 - 3.03iT - 31T^{2} \)
37 \( 1 + 3.03T + 37T^{2} \)
41 \( 1 - 0.464T + 41T^{2} \)
43 \( 1 + 11.3T + 43T^{2} \)
47 \( 1 - 0.928iT - 47T^{2} \)
53 \( 1 - 1.28T + 53T^{2} \)
59 \( 1 - 8.93iT - 59T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 + 6.54iT - 71T^{2} \)
73 \( 1 + 1.19iT - 73T^{2} \)
79 \( 1 - 9.40iT - 79T^{2} \)
83 \( 1 - 9.46iT - 83T^{2} \)
89 \( 1 + 11.7iT - 89T^{2} \)
97 \( 1 - 3.46iT - 97T^{2} \)
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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

−8.792564024043209593303350139045, −8.270476965902782863753006947441, −7.25336341015031217319217192157, −6.78433830343511216765706264549, −5.92239359106787808803085828882, −5.14879334010704281432759988141, −4.37635986313130270367953412938, −3.15646069034867160975867539471, −2.10589453400074996863966079956, −1.36440528369155428282432240105, 1.98442070371322511758463456712, 3.13471857283849355654494491261, 3.98468502670371873569942206532, 4.80527720741860364469053163289, 4.99981031336762681011484564586, 5.84252067230133671561682064913, 7.01746923723856604607979354977, 8.048438476139028192334400889702, 8.856117606221027547970008738153, 9.979947044087529302378376544142

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