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Properties

Label 1-35e2-1225.347-r1-0-0
Degree $1$
Conductor $1225$
Sign $-0.804 + 0.593i$
Analytic cond. $131.644$
Root an. cond. $131.644$
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  + (0.967 + 0.251i)2-s + (0.379 + 0.925i)3-s + (0.873 + 0.486i)4-s + (0.134 + 0.990i)6-s + (0.722 + 0.691i)8-s + (−0.712 + 0.701i)9-s + (0.712 + 0.701i)11-s + (−0.119 + 0.992i)12-s + (0.998 + 0.0448i)13-s + (0.525 + 0.850i)16-s + (0.999 + 0.0149i)17-s + (−0.866 + 0.5i)18-s + (−0.669 − 0.743i)19-s + (0.512 + 0.858i)22-s + (−0.800 + 0.599i)23-s + (−0.365 + 0.930i)24-s + ⋯
L(s)  = 1  + (0.967 + 0.251i)2-s + (0.379 + 0.925i)3-s + (0.873 + 0.486i)4-s + (0.134 + 0.990i)6-s + (0.722 + 0.691i)8-s + (−0.712 + 0.701i)9-s + (0.712 + 0.701i)11-s + (−0.119 + 0.992i)12-s + (0.998 + 0.0448i)13-s + (0.525 + 0.850i)16-s + (0.999 + 0.0149i)17-s + (−0.866 + 0.5i)18-s + (−0.669 − 0.743i)19-s + (0.512 + 0.858i)22-s + (−0.800 + 0.599i)23-s + (−0.365 + 0.930i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.804 + 0.593i$
Analytic conductor: \(131.644\)
Root analytic conductor: \(131.644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (1:\ ),\ -0.804 + 0.593i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.716526836 + 5.218340140i\)
\(L(\frac12)\) \(\approx\) \(1.716526836 + 5.218340140i\)
\(L(1)\) \(\approx\) \(1.893414118 + 1.470697687i\)
\(L(1)\) \(\approx\) \(1.893414118 + 1.470697687i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.967 + 0.251i)T \)
3 \( 1 + (0.379 + 0.925i)T \)
11 \( 1 + (0.712 + 0.701i)T \)
13 \( 1 + (0.998 + 0.0448i)T \)
17 \( 1 + (0.999 + 0.0149i)T \)
19 \( 1 + (-0.669 - 0.743i)T \)
23 \( 1 + (-0.800 + 0.599i)T \)
29 \( 1 + (0.995 - 0.0896i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (-0.119 + 0.992i)T \)
41 \( 1 + (0.983 - 0.178i)T \)
43 \( 1 + (0.433 + 0.900i)T \)
47 \( 1 + (-0.635 - 0.772i)T \)
53 \( 1 + (-0.486 + 0.873i)T \)
59 \( 1 + (0.999 - 0.0299i)T \)
61 \( 1 + (0.193 + 0.981i)T \)
67 \( 1 + (-0.207 - 0.978i)T \)
71 \( 1 + (0.858 - 0.512i)T \)
73 \( 1 + (0.538 + 0.842i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (-0.351 - 0.936i)T \)
89 \( 1 + (-0.887 - 0.460i)T \)
97 \( 1 + (-0.951 + 0.309i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.78016128275732844435133888006, −19.86905570736667094323362522885, −19.2017941215487682470066348282, −18.65406454402797594384164404843, −17.70353329727675403382624093794, −16.552752060959835012947971555599, −16.039197777752916132601676290075, −14.79294089752547201578958671667, −14.23640719335275856044245240741, −13.79960494167788965197323622202, −12.712635824425646059187298550730, −12.38114711125236864027799899171, −11.42680958968222801015093461424, −10.76005607885852283244286932082, −9.635340428347038849932064815350, −8.52110086618294329339326950336, −7.84468292145489143037626385350, −6.74962330579473873364544216292, −6.12519335070103770003494566409, −5.50895579769030098655309120959, −3.97118627222067223890152548703, −3.527613007771532956594542089322, −2.43942595409540397952323987219, −1.51933377304816963529896824890, −0.692028139448262658474492441303, 1.442798070020775631044340375751, 2.54454802156752766636054147754, 3.515104912379081697226984596573, 4.11731332923733758711416624430, 4.91645299296448385522689430098, 5.82784830665322265058236343305, 6.63789361691455592805328870083, 7.71793194261048680988676833226, 8.497362844410974445839209685578, 9.45218145074915992672612537663, 10.35675774802661154966591375517, 11.166958134363828637451525727234, 11.90010938840536887102537637247, 12.816517290939909834264330044, 13.74426739444145699028242532499, 14.3341776690846201326626779332, 15.039319336673857883612840423186, 15.71103490984956398318621596274, 16.38512602109096626772730878127, 17.099870732924901141020128411, 17.99022380433097059615789153979, 19.40790263881526644424625512548, 19.89982047546383485517339335686, 20.72507548477675112857294053792, 21.329068830039270526851842842612

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