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Properties

Label 1-35e2-1225.72-r1-0-0
Degree $1$
Conductor $1225$
Sign $-0.111 - 0.993i$
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  + (−0.701 + 0.712i)2-s + (0.611 − 0.791i)3-s + (−0.0149 − 0.999i)4-s + (0.134 + 0.990i)6-s + (0.722 + 0.691i)8-s + (−0.251 − 0.967i)9-s + (0.251 − 0.967i)11-s + (−0.800 − 0.599i)12-s + (0.998 + 0.0448i)13-s + (−0.999 + 0.0299i)16-s + (−0.486 − 0.873i)17-s + (0.866 + 0.5i)18-s + (0.978 − 0.207i)19-s + (0.512 + 0.858i)22-s + (−0.119 − 0.992i)23-s + (0.988 − 0.149i)24-s + ⋯
L(s)  = 1  + (−0.701 + 0.712i)2-s + (0.611 − 0.791i)3-s + (−0.0149 − 0.999i)4-s + (0.134 + 0.990i)6-s + (0.722 + 0.691i)8-s + (−0.251 − 0.967i)9-s + (0.251 − 0.967i)11-s + (−0.800 − 0.599i)12-s + (0.998 + 0.0448i)13-s + (−0.999 + 0.0299i)16-s + (−0.486 − 0.873i)17-s + (0.866 + 0.5i)18-s + (0.978 − 0.207i)19-s + (0.512 + 0.858i)22-s + (−0.119 − 0.992i)23-s + (0.988 − 0.149i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.111 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.111 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-0.111 - 0.993i$
Analytic conductor: \(131.644\)
Root analytic conductor: \(131.644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1225} (72, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1225,\ (1:\ ),\ -0.111 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.383451391 - 1.547749921i\)
\(L(\frac12)\) \(\approx\) \(1.383451391 - 1.547749921i\)
\(L(1)\) \(\approx\) \(1.022857271 - 0.2569805779i\)
\(L(1)\) \(\approx\) \(1.022857271 - 0.2569805779i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.701 + 0.712i)T \)
3 \( 1 + (0.611 - 0.791i)T \)
11 \( 1 + (0.251 - 0.967i)T \)
13 \( 1 + (0.998 + 0.0448i)T \)
17 \( 1 + (-0.486 - 0.873i)T \)
19 \( 1 + (0.978 - 0.207i)T \)
23 \( 1 + (-0.119 - 0.992i)T \)
29 \( 1 + (0.995 - 0.0896i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (-0.800 - 0.599i)T \)
41 \( 1 + (0.983 - 0.178i)T \)
43 \( 1 + (0.433 + 0.900i)T \)
47 \( 1 + (0.986 - 0.163i)T \)
53 \( 1 + (0.999 - 0.0149i)T \)
59 \( 1 + (-0.525 - 0.850i)T \)
61 \( 1 + (-0.946 - 0.323i)T \)
67 \( 1 + (-0.743 + 0.669i)T \)
71 \( 1 + (0.858 - 0.512i)T \)
73 \( 1 + (0.460 - 0.887i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (-0.351 - 0.936i)T \)
89 \( 1 + (0.842 - 0.538i)T \)
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

−20.99557102940991696005832610832, −20.318997676184679812148942680545, −19.7659251015582751556035023556, −19.05994159098761444354672809867, −18.10221708933679853515726564487, −17.41161678720967729214157382496, −16.64244719890415777221220421282, −15.62657865181251088476589134866, −15.348901234855696365187644868105, −13.966587467018670706900358545764, −13.4828807394636970708215397193, −12.41749788196107599471384949024, −11.61137737006842668372688305314, −10.72591643179220605041029403051, −10.11584076628603228035328586580, −9.36737976829252199765479565525, −8.69082713753942031537200809092, −7.91890626895658626060480694649, −7.09580392865085514948032833427, −5.79117786061835481482597938312, −4.49088542363014234935085652924, −3.89998053525107539502237326300, −3.034864554447213422660957962408, −2.05522340952143734061611979167, −1.14410042758472315902556425793, 0.58202193166375371376216050252, 1.11136605938490187101910344822, 2.368814047993610586281655410488, 3.31278054350825820637051995865, 4.61266579525533381711311329762, 5.831529078210990137247412535611, 6.44321290747302553052800434572, 7.21208180662296765366389489349, 8.06334720313627494083633951539, 8.80522790662414217599659838551, 9.21122028017529526987398953825, 10.42283691402455029419594187824, 11.26615890241709600220262833482, 12.10995341368889396433662329702, 13.31704543960414404982045336627, 14.02333784393581363790129576515, 14.27831442301595877661310012217, 15.663465717331877758560543564156, 15.94242647969516526339369783159, 16.96370886774086331540495347489, 18.00950243754111244570200322623, 18.23290337199638149003467033695, 19.118753192266724155760680430486, 19.72031856779850602236322432055, 20.46380384486146203269844079589

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